Stickfigure Emojis & Symbols

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣿⣿⣿⠿⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣦⣄⠀⠀⠀⠀⠀⠀⠙⢦⡀
⠀⠀⠀⣿⣇⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣹⣦⡀
⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⣿⣿⣷⣦⣤⣤⣤⣤⣶⣿⣿⣿⡿⡄
⠀⠀⠀⠈⣿⡇⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠟⠻⠿⣿⠾⢶⡿⢿⢿⡿⢿⡿⠷
⠀⠀⠀⠀⢻⣿⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠸⣿⡆⠀⠀⠀⠈⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠉⠉⠁
⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠛⢻⣿⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⢿⣿⠟⠛
⠀⠀⠀⠀⠀⢸⣿⡀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣧⣀⣀⣀⣀⣀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⢹⣿⣀⣀⣀⣀⣀⣈⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠈⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⡟⣿⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡇⣿⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠉⠉⠉⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⡇⣿⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⣿⣿⣿⣿⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢻⣿⣿⡟⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉


⡿⢿⠿⠿⡿⠻⠿⡿⠿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣻⣟⣻⣟⣻⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣴⣶⣿⣿⣿⣿⣿⣿⣿⣛⣋⣁⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣦⣤⣄⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡛⠛⠲⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠉⠉⠉⠛⠻⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣏⠛⠿⣿⣿⣟⠛⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠃⠀⠀⠙⠛⠦⠀⠀⠙⠻⣷⡄⠈⠙⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⡇
⢹⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⡇
⠀⢿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠃
⠀⠘⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⡆⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣯⡀
⠀⣾⣿⠟⠋⠛⠛⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠉⠙⢿⣿
⣼⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⢹⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠻⣿⣦⣤⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣤⣾⡿
⠀⠀⠈⠛⠛⠛⠿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣦⣤⣤⣤⣴⣿⣿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠛⠛⠋
⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠛⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⣿⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢿⣷⣦⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⡿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⢿⣷⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣶⣾⡿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⣿⣿⠿⣿⣿⣿⣿⡿⠿⣿⣿⠟⠛⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⣶⣶⣶⣶⣿⣿⣄⡀⠀⠀⠀⢀⣠⣿⣿⣶⣶⣶⣶⣦⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡟⠋⠉⠉⠉⠉⠉⠙⠻⣿⣷⣶⣾⣿⠿⠛⠁⠉⠉⠉⠉⠉⠻⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⡿⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⢹⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠁⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣷⡀⠀⠀⢿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⡏⠀⠀⢀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⠀⠀⠈⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣷⣦⣤⣼⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣤⣤⣶⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡏⠉⠙⢻⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠋⠉⠙⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⣾⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇⠀⠀⢹⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡟⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣇⠀⠀⠸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⢸⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠀⠀⠀⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⣾⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡆⠀⠀⢿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⠀⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢰⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⡿⠀⠀⢸⣿⣧⣿⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⣿⣯⣿⡇⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⠃⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠁⠀⠀⠈⢻⣿⡄
⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣨⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⠀⠀⠀⠈⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣧⠀⠀⠀⠀⢸⣿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⡀⠀⠀⠀⢰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣷⣄⣀⣠⣾⡿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠸⣿⣇⡀⣠⣴⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⠿⠟⠛⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡏⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠙⠻⠿⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⣠⣤⣤⣴⣶⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣦⣤⣄

⠀⠀⢀⣠⠆⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠐⡆⠐⠠⠀⠀⣀⠀⠄⢀⡀
⠀⠀⠈⠰⠐⠀⠀⠀⠀⠀⢈⣀⠂⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃⠈⠀⠀⠰⠁⠀⠈⠀⠁⡀⠃
⠀⠀⠀⠁⠀⠀⠀⠠⠜⢀⠀⠀⡀⢀⠀⠠⠀⠀⠀⠀⢀⠴⠒⠍⠚⠁⠀⠀⠀⢤⠐⠀⠀⠀⠀⠀⠖
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠉
⢀⡄⠀⠀⠀⠀⢀⡄⠀⣄⡀⢀⠘⠀⠀⠀⠐⠁⠀⠀⢀⡀⠀⡀⠶⠀⠀⢠⠀⢀⠀⠘⠃⠀⠴⠀⠀⠀⠱
⠈⠀⠀⠀⠘⠀⠀⠀⠀⠄⠀⠤⠂⠀⠀⠒⠠⡀⠰⠀⠀⠀⠈⠐⡀⠀⠀⠀⢀⠂⠐⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⡉⡁⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠐⠒⢀⡈⠒⠀⠀⠚⠀⠁⠀⠀⠈⠉⠀⠈⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠁⠈⢀⣀⠀⠀⠉⠀⠈
⠀⠀⠀⠀⠠⢉⠀⠀⢰⠋⠉⠓⠀⠀⠰⠤⢀⠀⠀⠀⠀⠀⡤⠠⠀⠀⠀⠀⠀⢦⠷⠸⠓⠃⠀⠠
⠀⠀⠀⠀⠁⠀⠀⠀⠀⢡⡔⠃⠀⠀⠀⠊⢀⠀⠀⠀⠠⠾⡰⢄⠗⠀⠀⠄⠀⠸⡰⠀⠢⠀⠐⠂
⠀⠀⠀⠚⢀⠀⠁⠀⠐⠂⠇⠉⠀⠀⠀⠋⠛⢂⣉
⠀⠀⠀⠀⠈⠄⠀⠀⠀⢐⢐⡁⠀⠀⠀⠚⠘⠈⠒⠀⠁⠀⠀⠐⠀⠁⠃⠀⠀⠀⠀⠈⠑⠂⠁⠊

⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⠀⠀⠀⠀⠀⠀⣠⣤⡀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⢀⡖⠀⣴⣶⠀⠲⡄⠀⠀⠀⠀⢰⣿⡷
⠀⣈⣟⣁⠀⠀⠀⠀⠀⠀⠀⠀⢻⠟⠀⠀⠀⠀⠀⣀⡻⠟⠁⠀⠀⠀⢀⣸⣿⣇⡀⠀⠀⠘⠧⣤⣽⣏⣤⠴⠃⠀⠀⠀⣰⣶⠉
⣼⣿⣿⣿⣷⠀⠀⠀⠀⠀⢀⠾⣿⣿⢦⠀⠀⢶⣻⣿⡇⠀⠀⠀⠀⢰⠏⣿⣿⣿⠹⡆⠀⠀⠀⢸⣿⣿⡇⠀⠀⠀⠀⠀⣿⣿
⡿⣿⣿⣿⢸⠀⠀⠿⠃⠀⢸⢀⣿⣿⠈⡇⠀⠈⢻⣿⡿⣦⠀⠀⠀⢸⡀⣿⣿⣿⢀⡇⠀⠀⠀⠐⣿⣿⡇⠀⠀⠀⠀⠀⢿⣿⣦
⠉⣿⣿⣿⠈⠀⣾⣿⣷⠀⠈⣼⣿⣿⣇⠁⠀⠀⠈⢻⣿⡟⠀⠀⠀⠀⢻⣿⣿⣿⡿⠀⠀⠀⠀⢸⣿⢹⡇⠀⠀⠀⠀⠀⠀⢻⡇
⠀⣿⣿⣿⠀⠀⢹⣿⡇⠀⠀⠛⣿⣿⠛⠀⠀⠀⠀⠈⣿⠇⠀⠀⠀⠀⠘⣿⣿⣿⠃⠀⠀⠀⠀⢸⡏⢸⣷⠀⠀⠀⠀⠀⠀⢸⡇
⠀⣿⣿⣿⠀⠀⢸⣿⡇⠀⠀⠀⣿⣿⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⣾⡇⠘⣿⠀⠀⠀⠀⠀⠀⠿⠃⢀⣀⡀
⠀⠉⠉⠉⠀⠀⠈⠉⠁⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠉⠁⠀⠉⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁


⠀⠀⠀⠀⢀⣶⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣄⠀⠀⠀⣰⣷⣀⡀⠀⠀⠀⣠⣤⣤
⠀⠀⣀⠀⣠⡟⠋⠀⠀⠀⢀⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣆⠀⠀⣿⡟⠛⠿⠿⣷⣶⣿⣿⣿⣷⣤⡀
⠀⠀⢿⢰⣿⡟⠒⠖⣶⣶⢘⣏⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⠀⠀⠀⠀⠀⢻⣧⠀⣿⠇⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣆
⠀⠀⢸⠸⣿⣧⣬⣭⡉⢹⡏⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⣴⣯⣜⠿⠃⠀⠀⠀⠀⠀⠀⠀⣹⣷⣿⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⣸⠖⠚⢿⢉⣿⠃⢸⡇⠀⠀⠀⠀⠀⠀⢠⣾⢷⣤⣤⣾⣿⣿⣇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠃⢻⡷
⠀⠀⠿⠉⠉⠻⠺⠏⠀⠘⠇⠀⠀⠀⠰⠶⠶⠿⠷⠶⠿⠿⠻⠯⠤⠿⠀⠀⠀⠀⠀⢀⣤⠴⠿⡿⠷⣄⠀⠀⠀⠀⣼⣿⡿⠛⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠋⠀⠰⣿⡿⠀⠘⡇⠀⠀⢠⣿⣿⠇⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⢸⣿⣿⠀⠀⠀⣿⣿⡇
⠀⠀⢀⣤⡀⠀⣠⣄⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠈⠉
⠀⠀⠹⡿⢋⣻⠿⡿⢉⣳⠆⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⣤⣤⡀
⠀⣴⣿⣿⣏⣿⣿⣿⡏⠁⠀⡇⠀⠀⠀⠀⠀⠐⠛⠋⠉⣻⣿⣶⣄⠀⠀⠀⢀⣀⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀
⢰⡇⣿⣿⣿⡇⣿⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠏⠈⠳⣦⣴⠋⣿⣿⠧⡤⠞⠀⠀⠀⠀⣼⣤⣤⣤⣤⣤⣾⣿⣧⢤⣤⣤⣤⣼⡇
⠘⠇⣿⣿⡿⠃⣿⣿⡇⠀⠀⡇⠀⠀⠀⠀⠀⠀⢀⣴⡿⣿⡟⠀⠀⠀⠈⠁⢠⣿⣏⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠙⠞⣿⣿⡿⠾⠃⠀⠀⠘⠃
⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⡇⠀⠀⠀⠀⢀⣴⡿⠋⢰⣿⠁⠀⠀⠀⠀⠀⠛⣿⠿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣿⣿⣧⣤⡄
⠀⠀⣿⣿⡇⠀⣿⣿⡇⠀⠀⡇⠀⠀⠀⠀⠛⠋⠀⢀⣿⠃⠀⠀⠀⠀⠀⢀⣾⠏⠀⢸⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣯⠉⠉⢉⣿⠃
⣤⣄⣿⣽⣧⣬⣭⣍⣡⣤⣤⣥⣤⣤⣤⣤⣤⣤⣤⣼⣯⣤⣤⣤⣤⣤⣤⣽⣥⣤⣤⣼⣯⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⣦⣤⣼⣯⣤⣤⣤⣤⣤
⣿⣿⣋⣉⣉⡉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠐⠀⠀⠀⠀⠀⠢⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⣀⠂⠀⠠
⠀⠀⠀⠰⠀⠀⠀⠃⠀⠀⠀⢀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠐⡀⠀⠀⠀⠀⠀⠀⠁⠐⠒⠀⠀⠇⡂⠀⠀⠀⠀⢠⠀⠀⡀⠀⠀⠀⠀⠁
⠀⠀⠀⠈⠀⢤⡈⠀⠀⢀⡀⠀⠀⠀⠀⠀⠑⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡤⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢀⠀⠂⠄⠐
⠀⠀⠀⠀⠀⠰⠘⠈⠁⢀⡵⠀⠀⠀⠀⠀⠀⢀⣤⠀⠃⠀⢀⠄⠀⠀⣤⠀⠀⠀⠀⠀⠀⠀⠀⠈⠩⠀⢘⡠⠀⠀⠀⠀⠾⠈⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢃⠐⠉⢠⡀⠀⠁⠀⠀⠀⣤⠀⠀⠀⠀⠀⠠⠠⠀⠁⠀⠀⠀⠢⠄⢀
⡠⢴⡆⢠⠀⠀⠀⠀⠉⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠄⠀⠐⠀⠀⠀⠘⠀⠸⠀⠀⠀⠀⠀⠐⠈⠉
⠁⠈⢁⢠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠐⠂⠀⡄⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠀⠀⠊⠀⣀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠈⠀⠀⣀⠀⢀⣃⠀⠁⠀⠀⠀⠸⡋⠉⠀⠀⠀⠀⠀⠀⠉⠆⠁⠈⠀⠉⠀⠀⠀⠀⠀⠀⠀⠑⠃⠀⠀⠀⠀⠐⠀⠃
⠀⠀⠀⠀⠈⠀⠀⠉⠀⠁⠀⢉⠀⠀⠀⠀⡀⣰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣱⠑⠓⠀⠀⠀⠀⠂⠔⠀⠀⠂⠀⠂⡐⠃⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠂⠀⠀⠀⠀⠀⠤⠸⣀⣀⣀⣀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⢈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⢀⠀⠀⠀⠀⠈⠒⢪⡀⠀⠀⠀⠀⠀⠀⠘⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠚⠀⠀⠀⠀⠀⣤⠀⠴⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠎⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⣶⢲⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢴⣻⡗⠀⢀
⠀⠀⠀⠀⠀⠀⢐⡋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠇⠀⠹⠃
⠀⠀⣀⢠⣶⣶⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣷⡄
⢀⣾⣿⢿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣻⣿⠇
⣼⣿⣯⣤⣍⣛⣩⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀⠀⠀⣠⣬⣭⣥⣤⡀⠀⢀⣀
⠉⠙⠛⣿⣿⣿⣿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣦⣾⣿⣿⣿⣿⣿⣿⣶⣿⡿
⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⣿⣿⣿⣿⣿⡿⠿⠟
⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇
⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇
⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇
⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇
⠀⠀⠀⢿⣿⡇⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⠇⢿⡿⠇

⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢾⣹⠇⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⢖⣶
⠀⠀⠀⠀⠀⠀⠀⠈⠆⠀⠝⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡟⠁
⠀⠀⠀⣾⣷⣠⣾⣿⣷⣄⣾⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣷⡄
⠀⠀⣰⣿⡟⢻⣿⣿⣿⡏⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣷⡄
⠀⠀⢿⣿⣷⣤⣭⣭⣥⣤⣾⣿⡿⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣬⣭⣭⣬⣿⣿⣄
⠀⠀⠀⠈⠙⣿⣿⣿⣿⣿⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⡿⠿⠿
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣟⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⡇⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⢻⣿⠇⢿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠟⢿⣿⠇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⣤⡤⠤⠤⠤⠤⣴⣶⣶⡶⠶⠤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠶⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢦⡀⠀⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠶⢤⣤⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⣀⣀⣸⣧⣤⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⢉⣩⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⡶⢶⡄⠀⠀⠀⠀⠀⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡶⣦⣾⠀⠙⠇⠀⣧⡴⣳⠄⠀⠀⣰⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢾⣿⣤⣤⣤⣤⣿⠟⠉⠀⢀⡴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣆⠀⠀⠀⣠⡟⠀⠀⣠⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠓⠒⠛⣉⣀⡴⠞⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢾⡇⣠⠀⠀⠀⠀⠀⢰⡞⠋⠉⠈⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠛⢿⣿⣟⣴⡞⠃⢀⡀⠸⣿⡀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⣰⢟⣿⣿⣷⣞⢻⡿⣼⡟⣇⠀⠀⠀⠀⢹⡄⠀⠀⣀⣤⣴⣦⣤⣀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡞⠙⠶⣤⡾⠛⠋⠉⢻⣿⣿⡟⠁⠈⠁⣿⠀⠀⠀⢀⣀⣷⠶⠛⠉⠙⣏⠻⢿⣿⣿⣶⣤⣤⣤⣾⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠶⡴⠿⠤⠤⠤⠶⢶⡟⠛⠛⠶⡶⣶⠿⠶⢶⡿⢻⡏⣿⠀⠀⠀⠀⢹⡆⠀⠁⠘⠏⢻⡿⣿⡿⠿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⢧⡻⣄⣠⠞⣁⡞⢡⡟⠀⠀⠀⣴⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠈⠳⢼⣟⣛⣩⡴⠋⠀⠀⠀⣰⠃⢀⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠋⠙⠛⢧⡀⠀⠀⠀⠀⠀⠀⠙⢯⣿⠀⠀⠀⣀⣰⡷⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⢹⡟⠲⠶⠶⠦⠴⠶⢶⣿⡷⠛⣿⠉⢹⡄⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠃⠀⠀⠀⠀⠘⣇⠀⠀⠀⠀⠀⠀⠋⠁⠀⠀⠘⢦⣄⣙⠛⠛⠉⢻⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⣆⠀⠀⠀⠀⠀⠹⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢩⡇⠀⠀⠀⣻⣷⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠈⠛⠲⢦⣴⡶⠀⠀⠀⠀⠀⠀⢀⣼⣧⡀⠀⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉

⡀⣀⡀⢀⡀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣷⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⣀⠀⠀⠀⣴⣦⠀⠀⠀⠐⣦⠀⠀⣴⣦⠀⠀⣴⠆⠀⠀⠀⢠⣦⡄
⠀⠀⠀⢠⣼⣿⣄⠀⠀⠀⠀⠀⣠⣽⣧⣄⠀⠀⠈⠻⠶⢦⣽⣯⣄⠀⠀⠀⠘⠳⢶⣽⣯⡴⠞⠋⠀⠀⠀⢀⣬⣿⣤
⠀⠀⢰⡟⣿⣿⡟⣧⠀⠀⠀⣼⢻⣿⣿⠛⠶⣤⠀⠀⠀⠀⣿⣿⡟⣧⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠠⠶⠶⠏⣿⣿⡟⣷
⠀⠀⠘⠃⣿⣿⡇⠛⠀⠀⠀⠛⢸⡿⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠛⠀⠀⠀⠀⠀⣿⢿⡇⠀⠀⠀⠀⠀⠀⣿⢿⣇⠛
⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⡗⢸⡇⠀⠀⠀⠀⠀⢠⣿⣿⡇⠀⠀⠀⠀⠀⢀⣿⢸⣷⠀⠀⠀⠀⠀⠀⣿⣸⣿
⠀⠀⠀⠀⠿⠿⠇⠀⠀⠀⠀⠀⠸⡟⠸⠇⠀⠀⠀⠀⠀⠸⠇⠿⠇⠀⠀⠀⠀⠀⠸⠇⠸⠿⠀⠀⠀⠀⠀⠀⠿⠿⠃

⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⣠⡄⢀⠀⣤⡀⠀⠀⡀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⢺⡿⠀⠀⠀⠀⠸⣯⣐⣿⣗⣨⡿⠂⠀⢻⣤⣀⢻⡿⠀⠀⠀⠀⠀⠀⠐⣿⠇⠀⠀⠀⠀⠀⠀⠸⣿⠃
⠀⢀⣴⠞⣿⣿⡟⢦⡄⠀⠀⠀⢹⣿⣿⠁⠀⠀⠀⠀⠀⠙⣿⣿⡟⢦⡄⠀⠀⢠⣿⣿⣿⣆⠀⠀⠀⠀⣰⣿⣿⣿⡀
⠀⠀⠙⠷⣿⣿⡷⠟⠁⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡷⠟⠁⠀⠀⠙⢿⣿⣿⠛⠀⠀⠰⠞⠋⣿⣿⡏⠙⠳
⠀⠀⠀⢠⣿⢸⣇⠀⠀⠀⠀⠀⣼⡇⢿⡆⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣸⡏⣿⠀⠀⠀⠀⠀⠀⢻⣿⡇
⠀⠀⠀⣸⡇⠈⣿⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⢰⡿⢹⣇⠀⠀⠀⠀⠀⢸⣷⣿⠀⠀⠀⠀⠀⠀⠘⣿⣄
⠀⠀⠀⠙⠁⠀⠉⠀⠀⠀⠀⠀⠉⠀⠈⠁⠀⠀⠀⠀⠀⠈⠁⠈⠋⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⣤⣦⡀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⠀⠀⠀⠀⠀⣠⣦⡀⠀⠀⠀⠀⠀⠀⣴⣦⠀⠀⠀⠀⠀⠀⢠⣶⡄
⠀⠀⠀⢠⣼⣯⣄⠀⠀⠀⠀⣀⣬⣿⣥⣄⠀⠀⠀⠀⠀⢠⣬⣿⣤⠀⠀⠀⠀⠀⣠⣽⣯⡄⠀⠀⠀⠀⢀⣬⣿⣤⡀
⠀⠀⠰⣿⣿⣿⣿⡧⠘⠛⠛⠋⢹⣿⣿⠉⠛⠛⠃⠀⠀⠸⣿⣿⡿⠀⠀⠀⣠⣴⢿⣿⣿⠃⠀⠀⢀⣴⠟⣿⣿⡏⠻⣄
⠀⠀⠀⠀⣿⢿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠉⠀⢸⣿⣿⠀⠀⠀⠈⠁⠀⣿⢿⣇⠀⠈
⠀⠀⠀⠀⣿⠸⣷⠀⠀⠀⠀⠀⠈⣿⣿⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⢸⣿⣿⠀⠀⠀⠀⠀⠀⣿⢸⣿
⠀⠀⠀⠀⠿⠀⢿⠀⠀⠀⠀⠀⠸⠿⡿⠀⠀⠀⠀⠀⠀⠸⠟⠘⢿⠀⠀⠀⠀⠀⠸⡿⠟⠀⠀⠀⠀⠀⠠⡿⠀⠿

⠀⠀⠀⢀⠀⢶⠀⣀⠀⠀⠀⢰⡆⠀⠀⠀⢴⣴⣶⠄⠀⠰⣤⡄⠀⠀⠀⢲⣤⣴⠂⠀⠀⠀⢰⠆⠀⠀⠀⠰⣄⠀⣰⠂⠀⠀⠀⣤⡴
⠀⠀⠀⠈⢻⣿⡟⠁⠀⠀⢴⣿⣿⡦⠀⠀⠈⣿⡏⠀⠀⠀⣿⣷⣦⠀⠀⠈⣿⡏⠀⠀⢠⠔⣿⣿⢦⠀⠀⠀⢹⣾⠁⠀⠀⢠⣶⣿⡅
⠀⠀⠀⠀⢸⡟⡇⠀⠀⠀⠀⣿⣿⠀⠀⠀⠀⡟⡇⠀⠀⠀⣿⢿⠁⠀⠀⠀⡿⡇⠀⠀⠀⠀⡟⣿⠀⠀⠀⠀⣸⣿⡄⠀⠀⠀⢹⢿⡇
⠀⠀⠀⠀⠘⠃⠃⠀⠀⠀⠀⠛⠛⠀⠀⠀⠀⠃⠃⠀⠀⠀⠛⠘⠀⠀⠀⠀⠃⠇⠀⠀⠀⠀⠃⠛⠀⠀⠀⠐⠋⠉⠙⠀⠀⠀⠸⠚⠃


⠀⠀⠀⠀⢲⣴⣴⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⠀⠀⠀⢴⠆⠀⠀⠀⠀⢀⣀⣰⣇⣀⣀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠰⡆
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⣶⣿⣿⡦⠀⠀⠀⠀⠉⠉⣿⡟⠉⠁⠀⠀⠈⠉⣿⣿⠉⠁⠀⠀⠀⠀⣠⣿⣤⡀⠀⠀⠀⢰⣿⣿⣷
⠀⠀⠀⠀⣸⠛⢧⠀⠀⠀⣴⡟⠛⠛⠛⠛⣷⡄⠀⠀⠀⣰⠋⠻⡄⠀⠀⠀⠀⠀⣿⢻⠀⠀⠀⠀⠠⠬⠿⠿⠿⠥⠄⠀⠀⢀⡞⠙⣆
⠀⠀⠀⠈⠁⠀⠈⠃⠀⠀⠈⠁⠀⠀⠀⠀⠉⠁⠀⠀⠀⠁⠀⠀⠉⠀⠀⠀⠀⠀⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈

⠀⠀⠀⢀
⠀⠀⠀⠈⢳⣻⡂⠀⠀⠀⠀⣸⣃⠀⠀⠀⠀⣘⣃⠀⠀⠀⠀⢐⣟⡼⠁⠀⠘⣆⣶⠀⠀⠀⠰⣿⣿⠆⠀⠀⠀⣶⡜⠀⠀⠀⢸⢸
⠀⠀⠀⠀⢸⣿⣿⠆⠀⠠⢾⣿⣿⠳⠀⠀⠺⣿⣿⡷⠀⠀⡴⣿⣿⡄⠀⠀⠀⢸⣿⡷⡄⠀⠀⣿⣿⠀⠀⠰⣿⣿⡄⠀⠀⠀⣾⣾⠆
⠀⠀⠀⠀⢸⡟⡟⠀⠀⠀⠈⡟⣿⠀⠀⠀⠀⣿⢻⠁⠀⠀⠀⢻⢻⠃⠀⠀⠀⢹⣿⠃⠀⠀⠀⣿⢻⠁⠀⠀⢹⢻⠃⠀⠀⠀⡼⡿⡄
⠀⠀⠀⠀⠘⠇⠇⠀⠀⠀⠀⠃⠛⠀⠀⠀⠀⠛⠚⠀⠀⠀⠀⠸⠸⠀⠀⠀⠀⠘⠛⠀⠀⠀⠀⠛⠘⠀⠀⠀⠘⠘⠀⠀⠀⠚⠈⠃⠙⠂


⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⡀⠀⠀⠀⠀⣆⢠⡄⡴⠀⠀⠀⡀⠀⢶⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢲⡠⡆⡰⠂
⠀⠀⠈⠉⢹⣿⣏⠉⠁⠀⠀⠀⠈⢧⣛⡼⠁⠀⠀⠀⠀⠈⣷⣾⠁⠀⠀⠀⠘⢶⣿⡷⠊⠀⠀⠀⠀⠀⣨⣯⡀⠀⠀⠀⠀⠀⣹⣿⠁
⠀⠀⠀⠀⢸⡟⡟⠀⠀⠀⠀⠀⠀⣸⣿⣇⠀⠀⠀⠀⠀⢰⡿⢿⡄⠀⠀⠀⠀⠘⡟⡟⠀⠀⠀⠀⣀⣘⣿⣿⣟⣀⡀⠀⠀⠀⢿⢿⠃
⠀⠀⠀⠀⠸⠇⠇⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠀⠀⠰⠋⠀⠀⠙⠆⠀⠀⠀⠀⠇⠇⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠀⠀⠀⠀⠸⠸


⣿⣿⠟⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⡠⠤⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⡀
⠀⠀⠀⠀⡈⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢊⠢⠀⠀⠀⠄⡀⠈⠆⠀⠀⠀⠀⠀⠀⠐⠁⠠⠀⠀⠁⠀⠀⠀⠀⠀⠔⢗⠴⡖⠉⡄
⠀⠀⠀⠀⠀⠂⠠⡄⠊⠀⠀⠠⠄⡄⠀⠀⠑⠠⠈⠂⣤⡀⠠⠂⠠⠄⠂⠀⠀⠀⢈⡸⢤⡤⠐⠁⠀⠀⠀⠀⠀⠑⠌⣩⠠⠂⠸⠆
⠀⠄⠐⡄⠠⠐⠁⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠈⠁⠀⠀⠀⠀⠆⠔⠀⡄⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑
⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠡⠀⠀⠀⠀⠀⠀⠀⠀⠟⣶
⠀⠀⠀⠀⠀⠀⠠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠈
⠀⠀⠀⠀⠀⣠⢁⠀⡄⠀⠀⠀⠀⠀⠀⠀⢀⡀⣀⠀⠀⢃⠀⡀⠀⠀⠀⠀⠀⡀⢀⡠⠀⠐⣀⠀⡀⠀⠀⠀⡀⢀⡌⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠁⠈⠀⠉⠀⠁⠀⠀⠀⠀⠀⠈⠀⠈⠀⠀⠈⠀⠈⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠁⠀⠁⠀⠀⠈⠀⠈⠀⠀⠀⠁⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⢀⠄⡄⠀⠀⠠⡀⠄⠀⠀⠀⠀⠀⠀⢀⡀⢀
⠀⠀⠀⠀⠀⠀⣠⠄⠀⠀⠠⣈⠀⠀⠀⠀⠀⢀⠀⠀⣁⡈⠁⠀⠀⠀⠀⠀⠒⢘⠄⠚⠁⠑⠃⠙⠀⠀⠀⠀⠠⠂⣄⠀⠀⢑
⠀⠀⠀⠀⠀⠁⠀⠠⠀⠁⠀⡀⠀⠀⠀⠀⠀⠈⠀⢀⣀⠠⠊⠀⠀⠀⠀⠀⢀⠠⠀⠁⠀⠁⠄⠐⠀⠀⠀⠀⠀⠀⠤⡤⠐
⠀⠀⠀⠀⠈⠂⡀⠤⡆⠁⠊⠀⠀⠀⠀⠀⠀⠀⠠⠂⠹⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠃⠀⠁⠀⠀⠀⠀⡀⠀⠀⡠⠛⠄⡀⣀⠢
⠀⠀⢀⠀⡔⠁⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣆⠀⡀⠈⠐⠐⢤⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠆⠡⠂⠈⠀⠃⠀⠀⠀⠁
⠀⠀⠑⣘⠀⠀⠀⠀⠼⠀⠐⢀⠠⡠⠀⠀⠀⠀⠤⠁⠉⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⢁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠠⠎⠊⠂⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⡄⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠐⠀⠀⠘⠂⠐⠂⠀⠀⠀⠒⠉⠃⠀⠀⠃⠑⠂⠀⠀⠀⠀⠀⠀⠘⠋⠛⠓
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠄⠀⠁⠀⡐⠀⠀⠀⠔⠈⠁⠀⠑⡀⠀⠀⠀⢀⠔⠂⠉⠀⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⠄⠀⠀⠀⠄⠂⠉⠀⢰⡂
⠀⠀⠠⡀⣀⣁⠄⠀⠀⠈⠄⡀⢀⣀⠔⠁⠀⠀⡀⠐⢄⠀⠀⠀⠐⢀⠴⠀⠀⠀⠀⠀⠀⢠⡀⠁⠀⠨⠀⠀⠐⢄⠀⠐⠂⠉⠒
⠀⠀⠀⠔⠙⠢⡀⠀⠀⡀⠀⡀⢘⠁⢂⠀⠀⠀⣅⠂⠤⢄⣓⠠⠂⠀⢀⠁⠀⠀⠀⠀⠐⠋⠩⠐⠂⠁⠀⠀⠀⠀⠀⢹⠀⠀⠈
⠀⠀⢰⠀⠠⠀⠀⠈⠀⠈⠀⠀⠈⢀⠸⠀⠀⠀⢢⠀⠀⠀⠈⠀⠀⠀⢈⠁⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⢅⡃⢰⠀⠀⠀⠀⠀⠀⠀⢠⠇⠤⠁⠀⠀⠀⠂⠄⢀⢀⡀⡀⠔⠁⠀⠀⠀⠀⢚⠔⢇⡐⠄⠀⠀⠀⠀⢐⢐⠀⢸
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠁⠀⠀⠀⠀⠀⠀⠀⠁⡈⠈⠒⠂⠀⠀⠀⠀⠀⠁
⠀⠀⠀⢠⠔⣲⡤⠀⠀⠀⠠⢴⠐⢴⠀⠀⠀⠀⢀⡄⣠⡄⠀⢸⣀⢠⡀⠀⠀⠀⠀⣴⠠⡀⢤⠀⣀⠀⠀⠀⠀⠀⠰⠐⠦⠠

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣇⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡶⠋⠉⠉⠛⢿⣧⣀⣤⡤
⠀⠀⠀⠀⠀⠀⢀⣤⠔⠚⠉⠙⠛⢷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⠹⣿⡅
⠀⠀⠀⠀⠀⢠⣾⠃⠀⠀⠀⠀⠀⠀⢻⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣾⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣄⠀⠀⠀⠀⢀⣼⠟
⠀⠀⠀⠀⢠⣿⣿⡄⠀⠀⠀⠀⠀⠀⢠⣿⠃⠛⠿⠶⠶⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢙⣷⡶⠶⢶⣿⠁
⠀⢀⡀⠀⣼⡟⠙⠿⣦⣄⣀⡀⣀⣠⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⠀⠀⢸⣿
⠀⠉⠛⠛⠋⠀⠀⣠⣾⣿⠋⠉⠹⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣄⡀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⣿⡇⠀⠀⣸⡿
⠀⠀⠀⠀⠀⢀⣾⠋⣾⡏⠀⠀⠀⢻⣿⠙⢿⣦⡀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣦⣴⣿⣿⣿⠀⠀⠀⠀⢀⣤⠾⠋⠀⢰⣿⠁⠀⣰⡿⢁⣾⡀
⠀⠀⠀⠀⣰⡿⠁⢰⣿⠁⢀⣤⠀⠈⣿⡇⠀⠙⢿⣦⠀⠀⠀⠀⠀⠈⢿⣿⣿⣿⣿⣿⠃⠀⣀⣤⠾⠛⠁⠀⠀⠀⠘⢋⣤⡾⠋⠀⠀⣿⣇
⠀⠀⠀⣰⡿⠀⠀⠚⣡⡴⠟⠁⢠⣤⣌⡛⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠈⠻⣿⣿⡿⠷⠞⠛⠉⠀⠀⠀⠀⢀⣀⣤⠾⠛⠁⠀⠀⠀⠀⢹⣿
⠀⠀⠀⣿⣇⣀⣴⠞⠋⠀⠀⠀⠀⠈⠙⠻⠷⣦⣤⣀⣿⠇⠀⠀⠀⠀⠀⠀⠈⠉⠀⢀⣀⣀⣠⣤⡴⠶⠛⠋⠁⢰⣾⠁⠀⠀⠀⠀⠀⠘⣿⡆
⠀⠀⠀⠈⠉⠁⣤⡖⠀⠀⠀⠀⠀⠀⠀⠸⣶⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⣼⣿⠀⠀⠀⠀⠀⠀⠀⣿⣇
⠀⠀⠀⠀⠀⢀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⠀⠀⠀⢹⣿
⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣄⣀⡀⢀⣀⣀⣀⣼⡿
⠀⠀⠀⠀⠀⣾⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡟⠛⠛⠋⣿⡏⠁
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢰⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢿⣷⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⠶⠶⠾⠛⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⣿⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⠁⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣾⠇⠀⠀⠀⠀⠀⣼⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠘⣿⡇
⠀⠀⠀⠀⠀⠀⠘⢿⣦⣀⠀⠀⠀⠀⠀⠛⠛⠷⠶⠦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠶⠶⠞⠛⠀⠀⠀⣀⣠⠿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡤⠒⠒⠒⠒⠒⠶⢤⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣊⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠒⣩⠋⣉⣩⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠊⠁⠀⣠⠗⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠜⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢠⠀⠀⠀⠀⠀⠘⠒⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠃⢠⣦⣄⠀⢰⣶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣆⡈⠻⠟⠀⠀⠛⠋⠀⠀⠀⠀⢀⣀⡤⠀⡠⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠈⠙⠦⣤⡤⠤⠤⠤⠶⠒⠚⠋⠉⣀⡤⠚⢧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⣠⣷⠈⠀⠀⠀⠀⠀⠉⠛⠒⠶⠶⠶⠒⠚⠉⠁⠀⡠⠊⢻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠴⠋⠀⠀⠀⢧⠙⢦
⠀⠀⠀⢀⣀⣀⡀⠀⠀⠀⠀⠀⢸⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠁⠀⠀⠀⠀⠀⠈⣇⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠛⢱⠋⠀⠉⢻⡟⠲⢦⣤⠤⢖⣋⣁⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⠘⢦
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⠄⠘⠦⣤⡤⠞⠁⠉⠉⠉⠉⠛⠛⠛⠛⠿⠿⠿⣶⣶⠶⠤⠤⠤⣤⣅⢀⣠⠼⠃
⠀⠀⠠⣤⡒⠂⠉⢁⡀⠀⠀⠒⠂⢠⣁⡀⠀⠀⠀⡀⠀⠀⠀⢀⣀⣀⣀⣀⡒⠉⢀⣀⣤⠤⠖⠊⠉⠀⠈⠉⠉⠛⠓⠒⠒⠲⠦⠤⠤⠤⣤
⠀⠀⠀⠀⠉⠙⠒⠦⢤⣀⡀⠀⠀⠤⠄⠀⡁⠀⠂⠀⠉⠀⠂⢿⣄⣀⣀⡤⠝⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠓⠒⠦⢤⣈⡙⠀⣀⠤⠒⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⢠⡶⢶⣄⡀⢀⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠙⠿⠀⠀⠈⢳⣟⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⣀⡴⢿⡁⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢼⡻⣇⠀⠀⠀⠀⠀⠀⢰⣟⣿⣧⡴⠾⠛⠉⠀⢸⡇⠀⠀⠀⠀⠀⠀⣾⣏⠉⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⢷⣿⡀⠀⠀⠀⠀⠀⠀⠉⠁⠹⣦⡀⠀⠀⢀⡼⠁⠀⠀⠀⠀⠀⣰⠿⠧⠶⠟
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⢸⡟⢦⡀⠀⠀⠀⠀⠀⠀⣀⣈⠙⠛⠛⢋⣀⡀⠀⠀⠀⠀⣴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⠀⠈⣧⠀⠙⢷⣄⠀⠀⣠⡼⢻⡏⠛⠛⠛⠋⠉⣿⣦⡀⢠⡾⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣶⠶⠟⠀⠀⠀⠉⠛⠛⠉⠀⣾⠀⠀⠀⠀⠀⠀⢻⡈⠛⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⢰⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣧⣀⣀⣀⣀⣀⣀⣀⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣯⠉⠉⠉⠉⠻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢷⡀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠃⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠏⠀⠀⠀⠀⠀⢀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣤⡾⠿⣶⠄⠀⠠⣦⣴⡿⠷⣶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣽⣛⡀⠻⢦⣤⠀⢠⣿⣁⡈⠳⣦⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⠛⠁⠀⠀⠉⠉⠛⠛⠋

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣏⣹⣏⣛⣛⣛⣫⣟⣛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡆⠀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⠀⠀⠟⡉⠉⠙⢶⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢀⠀⠀⢀⣠⠤⠤⢤⡀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠁⠀⠀⠘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠺⠁⠄⠀⠀⢉⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠂⢀⡁⠀⠂⠀⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⡄⢀⣡⠴⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠦⠤⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡄⣀⣀⣀⣀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⠤⠤⠄⠀⠀⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠒⠶⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣿⠠⠆⠂⢸⡇⠀⠀⠈⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠂⢨⠑⠀⠀⠀⠀⠜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠀⠐⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠃⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠜⠉⠉⡡
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠢⠀⠂⠠⠀⠀⢀⡄
⠀⠀⠀⠀⠀⡁⠀⠀⡴⠖⠑⢦⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠑⠒⢀⠆⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢈⡁⠀⠤⠒⠁
⠀⠀⠀⠀⠀⢃⠀⠀⡇⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠁⠂⠈⣠⡤⣉⠀⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠆⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠙⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⢀⡴⠶⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠈⠉⠉⠁⠂⠀⠀⠐⢀⠀⠀⢿⠀⠀⠀⢱⡄⠀⠀⠀⠀⠀⠀⠀⠀⠋⠁⠀⠙
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⢈⠂⡠⠀⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠠⡅⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠁⠠⡀⠀⠀⠀⠀⠀⢠⡄⠀⠀⠀⠀⠰⠤⠠⠄⠠⠤⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄⠀⠀⠀⠀⠀⠀⠀⢠⡤⠤⣤⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⠛⠛⠳⠄⠀⠀⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠈⡆⠀⠀⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠘⢁⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠀⠘⠀⠀⠀⠀⠀⠀⠀⠉⠉⠆⠙⠁⢀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠉⡟⠒⠂⢄⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⢢⠀⠀⠀⠀⠀⠀⠀⠀⠀⠔⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠒⠊⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠁⠀⡇⠀⠀⠀⠰⠀⠀⠀⠀⠀⠐⠃⠀⠀⠀⠐⠒⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠐⠂⠀⠀⡐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡧⠤⠠⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⢀⠁
⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⡂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠂⠈⠉⠉⠉⠁⠀⠈
⠀⠀⠀⠀⠀⠑⡀⠀⠀⠀⠠⠁⠐⠂⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄


⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣿⡿⠛⠿⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣥⣤⣤⣤⣤⣀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⠶⠶⠶⠶⠶⢤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣯⣀⣀⣀⣀⠀⢀⣀⣼⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠉⢉⣹⠟⠛⠛⠋⣽⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣀⠴⠋⠀⠀⠀⣠⠞⢹⣿⠀⠀⠀⢀⣤⠶⠖⠚⠛⠒⠶⠦⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡏⠀⠀⢀⡤⠞⠁⠀⢸⣿⠀⠀⠀⣿⣅⣀⠀⠀⠀⠀⠀⣀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠞⠋⠀⠀⠀⣠⢿⣿⠀⠀⠀⣿⠈⠉⠙⠛⢛⡿⠟⠛⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡴⠁⠀⠀⠀⣠⠞⠁⢸⡿⠀⠀⠀⣿⢀⣠⠶⠛⠉⠀⠀⣀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⢀⡴⠊⠁⠀⠀⣸⡇⠀⠀⠀⣿⠋⠀⠀⠀⣀⡴⠞⠉⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⡴⠋⠀⠀⠀⠀⡴⢻⡇⠀⠀⠀⡗⠀⣠⠔⠋⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠏⠀⠀⠀⢀⡴⠊⠀⢸⡇⠀⠀⠀⣧⠞⠁⠀⠀⠀⢀⡠⠞⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢀⣠⠖⠋⠀⠀⠀⣸⡇⠀⠀⠀⡏⠀⠀⢀⡤⠞⠉⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠋⠀⠀⠀⠀⢀⡴⢻⡇⠀⠀⠀⡇⣠⠞⠁⠀⠀⠀⢀⡰⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠃⠀⠀⠀⣀⠴⠋⠀⢸⡇⠀⠀⠀⣿⠁⠀⠀⠀⣀⠴⠋⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡠⠞⠁⠀⠀⠀⣼⡇⠀⠀⠀⡇⠀⢀⡴⠊⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣠⠊⠀⠀⠀⠀⣠⠞⣹⡇⠀⠀⢰⣇⡔⠉⠀⠀⠀⠀⣠⠜⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⢀⡴⠊⠁⠀⢿⡇⠀⠀⢸⡏⠀⠀⢀⣠⠖⠉⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣠⠖⠉⠀⠀⠀⢀⣿⡇⠀⠀⢸⡇⢠⠔⠉⠀⠀⠀⠀⣠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡜⠁⠀⠀⠀⢀⡴⠋⣼⡇⠀⠀⢸⡷⠃⠀⠀⠀⣀⠤⠊⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⢀⡤⠞⠁⠀⠀⣿⡇⠀⠀⢸⡇⠀⣀⠴⠊⠁⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⠔⠋⠀⠀⠀⠀⡠⣿⡇⠀⠀⢸⣧⠞⠁⠀⠀⠀⠀⣠⠞⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⠀⠀⠀⢀⡠⠚⠁⣿⡇⠀⠀⢸⡇⠀⠀⠀⣠⠴⠊⠁⠀⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣀⡤⠚⠉⠀⠀⠀⣿⡇⠀⠀⢸⠂⣠⠔⠋⠁⠀⠀⠀⢀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠞⠁⠀⠀⠀⠀⣠⠔⣿⡇⠀⠀⢸⡶⠁⠀⠀⠀⠀⢀⡤⠚⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣯⠀⠀⠀⣠⠔⠋⠁⠀⣿⠀⠀⠀⢸⡄⠀⠀⣀⠤⠚⠁⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⢀⡴⠋⠀⠀⠀⠀⠀⣿⡀⠀⠀⢸⢁⡤⠞⠁⠀⠀⠀⠀⢠⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣾⣄⣀⣀⣀⣀⣠⣤⡿⠁⠀⠀⢸⠏⠀⠀⠀⠀⣀⠴⠚⠉⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠉⠁⠀⠀⠀⠀⢼⠀⠀⢀⡴⠊⠁⠀⠀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣄⣰⠋⠀⠀⠀⢀⣜⣱⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠿⠿⠿⠿⠛⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠂⠈⠉⠉⠐⠒⠢⠄
⠀⠀⠀⠀⠀⠀⣀⢀⡀⠤⠤⢄⣀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠀⠀⠐⠀⠀⠠⢀⠀⠀⡀⠈⠀⡠⠤⠤⠒⠒⠢⢄⠀⠈⠤⡀
⠀⠀⠀⡠⠂⢁⠀⠀⠀⠀⠀⠀⠈⠑⢄⠀⠀⠀⢠⠖⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⡤⠊⠁⠀⠀⠀⠀⠀⠀⠀⠑⡄⠀⠈
⠀⠀⠈⢀⠔⠈⠉⢢⠀⠀⠀⠀⠀⠀⠀⢣⠀⡐⢁⠀⣀⣀⣠⣀⣀⡀⢆⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡄⠰
⠀⠠⠔⠁⠀⠀⠀⠀⠣⠠⠤⢀⡀⠀⠀⢸⢐⠀⡈⠉⠀⠀⠀⠀⠀⠈⢹⠀⠀⠀⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡧⡠
⠀⠀⠁⠀⠀⠀⠐⠀⠀⠀⠄⠀⠈⠑⢤⡜⠈⠀⠃⠀⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠀⣇⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⢰⢓⠅
⠀⠀⠈⢄⠀⠀⡀⠀⠀⠀⠀⠀⠀⢀⠞⠁⠠⠀⠘⡀⠀⠀⠀⠀⠀⡰⠃⠀⠀⠀⠀⢪⠦⣀⠀⠀⠄⢐⠈⠀⣀⠔⠁
⠘⠣⢄⣀⡑⠄⣈⡐⠒⠒⣁⡠⠔⠁⡠⠶⠆⠢⠠⠼⠤⡀⠀⠁⠘⢢⣤⣀⣀⣀⡠⠊⠀⠈⠙⢒⠶⠒⠶⣋⠁
⠀⠀⠀⠠⢠⠜⠁⠉⠉⠘⠉⢦⡈⡉⠀⢀⡦⠤⠤⡄⠀⠋⠉⠉⠋⠟⠐⢤⠤⠄⠒⡣⠤⠎⠀⠁⠐⠄⠀⠈⢆⡑⠒⠢⢤
⠀⠀⠀⠀⠌⠀⠀⠀⠀⠀⠀⠀⢂⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠘⡒⠁⠀⠘⠃⠀⠀⠈⠂⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠐⠀⠀⠀⠀⠀⠀⠡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢃
⠀⢀⡠⠬⢁⡀⠂⠠⣤⠤⠤⠄⠤⠆⠀⠀⠀⠀⡐⠀⠀⠀⠀⠀⠀⠀⠀⠐⠄⠀⠀⠀⠀⢇⡀⠀⠀⠄⠠⠤⠤⠤⢼⣀⡀
⠀⢀⣳⡀⠀⠀⠉⠈⠈⢂⡔⠐⠒⠂⠀⠀⠀⠚⠤⢄⣀⠀⠀⠀⠀⠀⠀⢀⢀⡢⠀⠀⠀⢰⣀⣀⣀⣀⣦⠤⠔⠊⢁⡞
⠀⠀⠀⠀⠀⠀⠀⠀⢠⠎⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠱⡖⠒⠊⠉⠈⠳⣄⡴⠀⡀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠘⠓⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠛⠒⠲⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣸⡀⠀⠀⠀⠀⠈⠁⠀⠉⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⡀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠁⠀⠀⠈⢉⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⠛⠫⡉⢢⡉⠢⡈⢳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⣮⡄⢄⠀⡄⠲⡈⠣⠑⢲⠤⡄⠀⣼⠖⠀⠀⠀⡠⢺⠋⢍⠒⠀⠀⠁⠀⠁⠘⣿⠃
⠀⠀⠀⠀⠀⠀⠀⢸⣸⢸⡀⠣⠘⠀⠁⠀⠄⠈⣏⣥⠾⠃⠀⠀⠀⠀⡔⡏⢀⡀⠀⠀⠀⠀⡀⠀⠀⢸⡱
⠀⠀⠀⠀⠀⠀⠀⢠⠿⡀⠀⠂⠀⠀⠀⢀⡄⠀⡏⢀⣀⠀⠀⠀⠀⠀⠈⢇⠀⠁⡀⠀⠀⢀⡁⠀⠀⡬
⠀⠀⠀⠀⠘⣷⣀⣈⡽⠳⡀⠀⠒⠒⠒⠉⢀⡴⠡⡋⠈⠃⢀⠐⠠⡆⠀⠈⠢⡀⠙⠂⠐⠊⠀⢀⠔⠁⢠⡤⢤
⠀⠀⠀⠀⠀⣈⠉⠁⠀⠀⠈⢓⢶⡴⠦⣚⣉⣠⠔⠉⠁⠀⠈⠓⠚⠳⢄⣠⢤⡌⠓⡴⢤⣤⠖⠥⣤⠴⠚⢦⡼
⠀⠀⠀⠀⠸⣉⡹⠒⠒⠾⡉⠉⠀⠀⠀⡔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠤⢤⠀⠀⠀⠀⠴⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠁⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⡰⠁⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⣀⣀⣀⣀⠠⢧
⠀⠀⠀⠀⠀⠀⠀⢜⡁⠀⠀⠀⠀⠀⠀⢀⡵⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⣠⡀⠀⢀⡱⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣓⠒⠒⠒⠲⡏⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠓⠖⠒⠃⠑⠘⡀⣠⣻⡄⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣹⠀⠀⠀⣠⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⠀⠀⠀⠀⠈⠦⠠⠃
⠀⠀⠀⠀⠀⠀⠰⢏⣈⡹⠃⠀⠰⣛⣃⣲⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠒⠋⢣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⠉⠁





⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⣠⣤⣤⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀
⠀⣿⠀⠀⠐⠋⠉⠉⠉⠙⠛⠛⠛⠛⠒⠶⠶⠶⠶⠤⣤⣤⣤⣤⣤⣤⠤⠤⠴⠶⠶⠖⠚⠛⣉⣽⠇
⠀⢹⣆⣘⡒⠂⠠⠀⠀⠈⠉⠒⠒⠒⠒⠦⠀⠈⠉⠓⠂⠀⠀⠀⠀⡀⢀⡀⠠⠤⣤⢄⣀⠀⠙⠻⠶⠶⠤⠤⠤⡤⠤⠤⠶⠶⠶⠶⣆
⠀⠀⠈⠉⠉⠉⠛⠛⠛⠒⠲⠶⠦⣤⣄⣀⠀⠀⠀⠀⢠⡦⠤⣤⣤⣤⣤⣭⣉⣓⣓⣒⣛⣁⣀⣀⣙⠛⠒⠀⠐⠀⠀⠀⢶⣶⣶⡚⠛
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠲⣤⣀⠀⢳⣄⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠛⠛⠛⠲⠶⢦⣤⣽⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⡴⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠌⢓⡦⢄⡠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⢿⣶⣶⣦⣄⣀⠀⠀⠀⠀⠀⠀⠀⣼⣷⣾⣿⣿⣧⡀⣍
⠀⠀⠀⠀⠀⠀⠀⠀⣰⠋⠀⠀⠻⣿⣿⣿⣿⣿⣷⠖⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⡍
⠀⠀⠀⠀⠀⠀⠀⡼⠁⠀⠀⠀⠀⠈⢿⣿⡿⠋⠁⠀⠀⠀⠀⢻⡋⡻⠿⣿⣿⢿⣿⡇
⠀⠀⠀⠀⠀⢀⡞⠁⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣾⠅⡩⠗⠀⠖⢬⠹⢄
⠀⠀⠀⠀⣠⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣧⣾⣿⢦⡠⡬⠌⢒⣣⡀⠈⢻
⠀⠀⢠⣴⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⠁⠀⢰⢠⣥⣄⠍⠉⢘⠣⣤⡀⠑⢄
⠀⠀⣿⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡞⠻⣿⣿⡿⠀⠀⠈⢓⡿⡟⠿⠔⠁⢰⣿⣿⡄⠀⠑⢄
⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⢠⣒⢅⠀⢐⣤⠜⢁⡷⢦⣄⣠⣞⠀⡇⠀⠀⠀⣼⢿⡟⢹⡀⠀⠀⢳
⣀⣶⣿⣷⣆⣀⣀⣀⣀⣀⣀⡀⡋⠀⣩⣟⣶⡤⠤⢮⣷⣀⢀⡾⢀⠉⣿⠙⢹⣉⡿⠀⠙⠟⢼⡀⠀⠀⡇
⠻⠿⣏⣉⣙⣛⡛⠛⠛⠛⠛⡱⠚⡾⠿⠿⠿⠿⣿⣷⣶⣦⣭⣝⣒⠦⡼⠦⣞⣓⣀⣡⣭⣭⣭⣭⣿⣶⣥⠤⠤⠤⠤
⣀⣀⣈⣉⣉⣉⣉⣉⣉⣛⣛⣒⣈⣉⣉⣉⣉⣉⣉⣉⣛⣛⣛⣛⣛⣻⣧⣟⣛⣛⣓⣒⣒⣒⣒⣛⣛⣛⣛⣃⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⣠⡄⣶⢦⣀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⢀⢠⣦⣀⢄⡄
⠀⠀⠀⠀⢈⡟⠓⠙⠛⢷⣷⠁⠀⡰⠋⡹⠟⠟⠛⠳⢽⠯⡁⢱⠀⢀⡤⠞⣫⢶⣤⡘⠙⣶⢶⣶⢿⢿⣿⢿⣧⣤⣀
⠀⠀⠀⢀⡎⠀⠀⠈⠁⠀⢸⡀⠀⡇⣼⠁⡆⠀⠀⠀⡀⣇⢇⠇⠘⡏⢠⠐⠡⠀⠉⢻⢺⣿⡗⢁⠀⠀⠋⡀⢿⣿⣿
⠀⠀⠀⠙⢇⠘⢄⣀⡤⠂⡼⠋⢠⡷⠉⢆⠈⠒⠒⠊⣠⢿⠘⡄⠀⢣⡘⢤⠤⠂⢀⡼⠂⠛⣇⠘⠤⣀⠴⠁⡎⠹⡍
⠀⠀⠀⠀⢀⣱⠦⠤⢴⠊⠀⠉⠉⠀⠀⣀⣵⠒⠒⢎⠁⠀⠣⠝⢀⣠⡼⠒⢲⣒⡋⠀⢰⠷⠈⣲⠤⠤⢔⠊
⠀⠀⣀⢴⠉⡜⢦⡔⢺⢱⡄⠀⠰⣶⠊⢡⠟⢼⡀⢿⠣⡠⡦⠐⣇⡴⠁⠀⢸⣀⡟⠊⣁⣀⠼⡧⢖⠦⡜⢄⡀
⠘⠛⠀⢘⡟⠀⠀⠀⠈⡏⠈⢢⡄⠀⢠⠋⠠⠀⠀⠸⡀⠹⠖⠠⡎⠀⠀⠀⠸⠀⠁⠀⠟⣠⠊⠀⠀⠀⢰⡀⠸⠓
⠀⠀⠀⠀⣱⠲⠄⠤⢤⠃⠀⠈⠀⠀⠀⡽⠒⡶⡄⢴⠓⠀⠀⠀⡜⠉⠀⠀⢸⠁⠀⠀⠀⠙⠿⢖⠶⣲⡿⠵
⠀⠀⠀⣠⠃⠀⠔⡄⠘⡄⠀⠀⠀⠀⢠⠃⡸⠀⢧⠘⡄⠀⠀⠘⢦⠴⠓⠤⡜⠂⠀⠀⠀⠀⡤⣇⠀⠀⣣⢄
⠀⣀⠼⠤⣀⡞⠀⣟⣀⣸⡀⠀⠠⣊⣉⡻⠁⠀⠸⡟⠓⣄⣠⣒⣳⠀⠀⠀⣱⠒⣢⠀⢤⣚⣙⠇⠀⠀⠙⣉⣲⠄

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⠴⠿⠉⠉⠉⠉⠙⠓⠲⠦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⡋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⠀⠀⠀⠀⣠⠴⢒⡟⣀⣀⡤⠴⢒⠞⠀⣀⠴⠆⠀⠀⠀⠀⣸⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⡴⠋⠀⣀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠀⠸⡀⠀⠀⠀⠀⢻
⠀⠀⣤⣠⣄⠀⠀⠀⠀⢿⠀⢸⠀⠀⠉⠀⠈⠀⠀⠀⠀⠀⠀⠒⠢⢄⠀⢧⡄⠀⠀⠀⢸⠀⠀⣠⣄⢤
⣐⣿⢧⠿⢸⢾⡆⠀⠀⢸⢋⣾⠀⠀⠀⢰⣷⠀⠀⠀⠀⠀⠀⣠⣄⠀⠀⠈⢻⠀⣀⡀⡼⠰⡷⡇⠟⡼⣻
⠙⠻⡄⠀⠎⢸⠀⠀⠀⠸⡄⠛⠀⠀⠀⠈⠛⠀⡖⠃⠀⠀⠀⠻⠟⠀⠀⠀⠀⠛⡕⡿⠁⠀⡇⠱⠀⢠⠟
⠀⠀⠙⣦⠀⠸⡄⠀⠀⠀⠑⢾⠀⠀⠀⠀⣀⡀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⢠⠇⠀⣴⠋
⠀⠀⠀⠘⡆⠀⠙⣆⠀⠀⠀⠘⣆⠀⠀⠀⠸⡀⠉⠉⠉⠉⡟⠀⠀⠀⠀⠀⢠⠗⠃⠀⣰⠃⠀⢰⠃
⠀⠀⠀⠀⠘⣄⠀⠈⠳⢄⡀⠀⠘⢦⡀⠀⠀⠙⠢⠤⠔⠊⠀⠀⠀⠀⠀⡴⠃⢀⡠⠞⠁⠀⣠⠃
⠀⠀⠀⠀⠀⠘⢦⠀⠀⣠⣿⣷⣦⣤⣽⣦⣄⡀⠀⠀⠀⠀⠀⠀⣀⣴⣮⣴⣾⣿⣄⠀⠀⡴⠃
⠀⠀⠀⠀⠀⠀⠀⠑⣴⣿⣿⣿⣿⣿⣿⠟⠉⢍⠑⠒⠒⠒⠈⡟⠙⢿⣿⣿⣿⣿⣿⣦⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⠏⠀⠀⠈⠑⠠⠤⠐⠋⠀⠀⠈⣿⣿⣿⡿⠟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢻⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠧⣄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣷⣶⣶⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⠏⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⣠⣴⣾⠿⠿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⠿⠿⣻⣦⣄
⠀⠀⠀⠀⠀⠘⣇⠀⠀⠉⠒⠬⡙⢿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡿⢟⡥⠒⠉⠁⠀⣠⠇
⠀⠀⠀⠀⠀⠀⠀⠙⠲⢤⣀⡀⠈⢲⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣗⠁⠀⣀⡤⠖⠋⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠤⢤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠿⠀⠀⠀⠀⣠⠔⠒⠋⠀⠀⠀⠀⠀⠉⠓⢄⠀⠀⠀⠀⠀⢀⡤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⢀⠞⢀⡠⠶⣄⠀⠀⡀⠀⠀⠀⠀⠈⢻⣿⡦⣄⠀⠛⠒⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⠬⠓⠀⠀⡞⢀⡞⠀⠀⠘⢦⠀⢻⣆⡀⠀⠀⠀⠀⢿⠃⠈⢇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠀⠀⠀⠀⠙⠦⣳⡙⢦⡀⠀⠀⢸⡄⠀⡞
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡾⡀⠀⠀⠀⠀⠀⠀⠈⣉⠀⠈⢳⠶⣼⠇⢠⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡇⠀⠛⠀⠀⢀⠀⠀⠛⠀⠀⢸⣴⠿⣄⠀⠱⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⡀⠀⠀⠀⠤⠄⠀⠀⠀⣠⠟⠁⠀⠈⢳⢀⡇
⠀⠀⠀⣀⡤⠤⠤⣀⣀⡤⢤⣀⡀⠀⠀⠀⢠⠎⠈⠓⠦⢤⣀⣠⡤⠔⠋⠀⠉⢢⡀⠀⠼⠋⢀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⢾⡡⢖⣋⣑⣦⣥⣖⣲⣦⣍⡱⠀⠀⡏⠀⠀⠀⠀⠀⠙⠊⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⣯⣀⣀⣀⣀⣀⣀⣸
⠀⠀⠀⠀⣇⣀⣀⣀⣀⣀⣀⣸⣀⡀⠀⢸⠁⣶⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠘⡦⠤⠤⠤⠤⠤⠤⢼
⠀⠀⠀⠀⠈⢹⠁⠀⠀⠀⠀⠀⠀⡇⠀⢸⡞⠀⠭⣧⠀⠀⠀⠀⠀⠀⠀⢀⡤⢤⣀⡇⠀⢰⠋⠉⠉⠉⠉⠉⠉⢹
⠀⠀⠀⢠⠼⠭⠭⠭⠭⠭⠭⠭⠭⢇⠀⠸⣷⢌⡽⢛⣏⣉⠑⠲⠒⠋⠙⣧⢂⡄⢈⡇⠀⢸⣄⣀⣀⣀⣀⣀⣀⣀⡇
⠀⠀⠒⠓⠲⠶⠶⠶⠶⠶⠖⠲⠶⠚⠒⠒⠛⠻⣨⠴⠒⠒⢬⣒⣪⠥⠤⠬⢽⡶⠛⠒⠒⠺⠤⠤⠤⠤⠤⠤⠤⠤⠗⠒⠒

⠀⠀⠀⠀⡀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀
⠀⠀⠀⣀⣧⠀⠀⠀⢳⡇⠀⠀⠀⠀⢻⣧⣄⠀⠈⢿⡧⠤⠀⠀⠀⠀⣸⠀⠀⠀⢠⣿⡄⠀⠀⠀⣤⣯⡄⠀⠀⢠⣿⢆⠀⠀⠀⣠⣧⠗⠀⠀⠀⣬⣥⡀
⠀⠀⠀⢠⠟⠀⠀⠀⣾⣭⠁⠀⠠⠖⣹⠇⠈⠀⠀⢸⡇⠀⠀⠀⠀⣨⠿⠀⠀⠀⠘⣿⡟⠀⠀⠀⢹⢻⠁⠀⠀⢈⣿⠀⠀⠀⠀⢿⣿⠀⠀⠀⠀⢿⢧
⠀⠀⠀⠈⠂⠀⠀⠀⠚⠃⠀⠀⠀⠀⠃⠀⠀⠀⠀⠸⠇⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠛⠃⠀⠀⠀⠈⠘⠀⠀⠀⠈⠀⠃⠀⠀⠀⠘⠛⠀⠀⠀⠀⠘⠈
⠀⠀⠀⣀⠀⠀⠀⣄⢠⡄⢀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⣠⠀⠀⠀⠀⠄⢀⡄⠀⠀⡀⢠⡀⣀⠀⠀⣠⣀⢄⠀⠀⢠⡀
⠀⠀⢀⣬⣷⡤⠀⠈⢻⡿⠁⠀⠀⢩⣿⠯⠤⠀⠀⣻⣠⠄⠀⠀⠀⣾⠴⠂⠀⠐⠒⣶⡖⠒⠀⠀⢸⣿⡇⠀⠀⠈⣷⡞⠁⠀⠀⠑⣷⠎⠀⠀⢠⢻⣯⠃
⠀⠘⢫⠏⠀⠳⠀⢠⠋⠙⠤⠄⠀⡼⠙⡆⠀⠀⠀⠻⠚⠃⠀⠀⠈⢻⠆⠀⠀⠀⢀⡏⢇⠀⠀⠀⢸⢻⠁⠀⠀⠀⡏⢻⠀⠀⠀⠀⠋⡇⠀⠀⠀⡞⠹⡆
⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠁⠀⠀⠀⠁
⠀⠀⢰⠠⠄⠀⠀⠀⣠⣄⠖⠀⢀⡀⢶⣀⡀⠀⠀⢀⣶⠀⠀⠀⢰⡀⡄⠀⠀⠀⠀⣶⣠⠀⠀⠀⠀⣦⠀⠀⠀⢀⣠⡆⢀⠀⠀⠀⡶⠀⠀⠀⠀⣦⡄
⠀⠀⠀⣿⣿⠀⠀⠀⣱⢿⠷⠀⠀⠉⢹⣿⠈⠀⠀⣿⣿⡑⠀⠀⠈⢹⡟⠃⠀⠀⠌⣿⠃⠀⠀⠀⢯⣿⡿⠀⠈⠉⣿⡋⠁⠀⠀⠸⣿⡇⠀⠀⠀⢹⣿
⠀⠀⠀⡗⡇⠀⠀⢠⠇⠈⠆⠀⠀⠀⠹⠇⠀⠀⠀⠈⠇⠃⠀⠀⠀⢸⢻⠀⠀⠀⠰⠿⠄⠀⠀⠀⠸⠸⠀⠀⠀⠀⠿⠇⠀⠀⠀⠀⠟⠀⠀⠀⠀⢸⡏

⠀⠀⢽⡦⠞⠀⠀⠀⣬⡷⠂⠀⢀⡀⡆⠀⠀⠀⠀⠰⣿⡄⠀⠀⣀⣻⡴⠄⠀⠐⠦⣼⡴⠋⠀⢀⣿⣄⡀⠀⠀⢀⢴⣿⡄⠀⠠⢤⣯⠤⠄⠀⠀⠦⣼⡇
⠀⠀⢸⡟⠙⠄⠀⠐⠙⡗⢆⠀⠈⡽⢷⣾⣍⠀⠀⡴⢻⠈⠀⠀⠀⡴⢷⠀⠀⠀⢀⡿⢧⠀⠀⠘⣿⠀⠀⠀⠀⠀⣸⡿⠃⠀⠀⠀⣿⠀⠀⠀⠰⠛⢿
⠀⠀⠸⠀⠀⠀⠀⠀⠐⠃⠈⠀⠀⠙⠀⠀⠀⠀⠀⠁⠐⠀⠀⠀⠀⠀⠈⠓⠀⠘⠉⠀⠀⠳⠀⠀⠛⠀⠀⠀⠀⠈⠐⠃⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠘⠆
⠀⠀⠀⠀⢀⠀⠀⠀⣀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⢀⡀⠀⠀⠀⢀⠀⠀⠀⠀⠀⢀⣀⡄⠀⠀⠀⠀⣀⠀⠀⠀⢠⣤⡀⠀⠀⠀⠀⣀⢠
⠀⢀⣀⣣⣞⠀⠀⠀⡸⣿⡁⠀⠀⢀⣾⣦⡀⠀⠠⠴⣿⠖⠂⠀⠑⢮⡥⠤⠀⠀⢘⣶⣃⡀⠀⠀⢡⡿⡀⠀⠀⠐⢹⠧⠖⠀⢠⢾⣟⠁⠀⠀⠒⠒⣿⠋
⠀⠀⠀⣠⢻⡇⠀⠀⢠⢾⠁⠀⠀⢠⠟⢳⠀⠀⠀⢠⠛⢦⠀⠀⢀⡼⠳⡄⠀⠀⡟⢧⠀⠀⠀⠀⢹⢦⠀⠀⠀⢀⡟⢻⠀⠀⠈⠈⡏⢳⡀⠀⠀⠀⣿⡄
⠀⠀⠀⠘⠀⠑⠀⠀⠈⠈⠀⠀⠀⠈⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠁⠀⠘⠁⠘⠀⠀⠀⠀⠁⠈⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠃⠀⠁⠀⠀⠀⠛⠃
⣶⣶⣶⢶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣧⣤⣤⣤⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣤⣿⣿⣷⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡿⠛⠛⠿⠿⠿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣿⠁⢀⣄⠀⢀⣀⠘⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠋⠀⠈⠁⠀⠀⠋⠘⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⠀⠀⢀⣀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡴⠎⠉⠠⣀⠀⠈⢀⡀⠛⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣄⣀⣠⡈⠀⠀⠀⢂⠀⠀⠛⠀⣤⠀⠈⠻⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠏⢠⠀⠀⠈⠛⠳⢦⡀⢋⣱⣄⣠⣾⡏⣠⠤⠶⠈⣉⣠⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠁⠀⠀⠀⠀⠈⢻⡎⣛⡟⢛⣿⢀⡏⠀⠀⠀⠉⠋⠨⠙⢦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠳⣄⠀⠀⠀⠀⠀⠀⠁⢨⣟⠨⣯⠸⠇⠀⠀⠀⠀⠀⠀⠀⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⢘⣷⣄⠀⠀⠀⠀⠀⠬⣽⠆⣿⡆⠀⠀⠀⠀⠀⠀⠀⣼⠃⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠆⠀⠀⠀⠀⢸⡏⠹⣧⠀⠀⠀⢠⢞⣷⠖⣻⣿⢀⠀⠀⠀⠀⠀⢰⡟⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⠀⠀⠀⠀⡼⠁⠀⢻⠐⠂⠀⢸⡯⣟⡃⣿⡟⠈⠀⠀⠀⠀⠓⡼⠃⢠⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇⠀⠀⢸⡧⠞⣁⣿⣇⣀⠀⠀⠀⠀⠀⡇⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⢠⡇⠀⠀⠀⠁⠀⠀⢸⣷⠻⣭⡿⠉⠙⠀⠀⠠⠄⣴⠀⠀⢠⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⢵⣶⣿⠁⠀⠀⠀⢠⡆⠀⢸⣷⠛⣶⡃⠀⠸⠀⠀⠀⢰⡏⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠈⣿⠀⢠⣄⠀⠸⡇⠀⢸⣿⢽⣼⡇⠀⢀⡀⠀⠀⢸⡗⢦⠄⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⣀⣿⠀⠠⠁⠀⠴⣷⠀⢸⣿⢺⣼⣇⠀⠈⠁⠀⠒⡾⠋⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠹⠏⠀⠀⠀⠀⠀⠈⡀⢸⣿⣿⣺⣿⠀⠚⣧⠀⢀⣷⠀⠀⠀⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⣀⠁⠀⠀⠀⠀⠀⠀⢸⣟⢿⣿⡇⠀⠤⡿⠀⣸⢿⠀⠀⠀⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡏⢠⠀⠒⠀⣿⠀⠀⠀⠐⠁⠀⠀⣾⣯⡸⣾⣷⠀⠀⣀⠀⠙⣸⡀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢧⣤⡶⠶⠶⠾⡂⠀⠀⠀⠀⠀⠀⣿⠈⡍⠛⢉⣷⡻⠟⠀⠀⣻⡇⠀⠐⠀⢬⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⢡⠀⠀⠀⠀⠀⠀⣸⠐⢆⣽⠀⣿⣛⣾⣧⢀⡿⠒⠒⠲⠶⣶⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢠⣤⣌⣁⣀⣈⣉⣽⣶⣿⣿⣿⣿⣿⣿⣿⡿⠃⢰⡆⠀⠀⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠘⠓⢠⣸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⢠⠟⢀⣀⡶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠄⠐⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⡏⠀⢿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠃⠀⢸⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢾⣿⣿⣿⣿⣿⣿⡿⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⠇⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡏⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣿⠁⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠟⠻⠿⣿⣿⠄⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⣿⡷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠠⠀⢰⣿⣭⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣽⣿⣍⡀⠂⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣇⣠⡀⢰⣿⡟⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⣿⣿⣿⣿⣿⣤⡁⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠿⠿⣿⣿⡿⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠌⠉⠻⢿⣿⣷⣿⣿⠦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠂
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⣿⣿⣿⡏⠁⠈⠉⠉⠉⠉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣾⣷⣤⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠈⢷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣇⣀⣀⣀⣸⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣟⠛⠻⣷⣶⣤⡘⠀⠀⠀⠀⠀⠘⠲⣿⡛⢻⡿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⠿⣿⣿⣷⣜⠿⣿⣷⣿⣧⡀⠀⠀⠀⠀⠀⠀⠙⢾⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣶⣶⡄⠙⠟⣻⣿⣴⣿⣿⣿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣷⣿⣿⣿⣾⡟⠋⠛⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⢻⠩⣍⠛⠿⠃⠀⠄⢤⠆⠀⠀⠀⠀⠀⣤⠀⠀⠀⡿
⠠⠤⠤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡼⠉⠀⠀⠈⢎⡓⠀⠀⠀⠙⠋⠁⠀⠀⠀⠀⡀⠉⡉⠉⢏⣽⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠞⠻⣤⢤⡀⠀⠀⠀⠀⠀⠀⠀⣀⣐⠒⠀⠀⠀⢁⣁⣠⡴⢚⣩⣼⠀⠀⠀⠐⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠔⠯⣄⡀⠈⠳⡝⢦⡀⠀⢄⡀⠀⠀⠀⠌⠀⠀⣉⣀⢚⣋⣶⣶⣾⣿⡯
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠃⠀⠀⠀⣙⣦⣀⡙⣆⠙⠢⠄⠁⠈⠡⠦⠅⠂⢀⡻⠷⠈⢘⠁
⠀⠀⠀⠀⠀⠀⠀⣠⠞⠁⡠⠤⠤⢄⣈⠑⢦⣹⠈⢧⣀⢻⢀⠀⠀⠀⢀⠀⠈⠁⠀⠀⠸
⠀⠀⠀⠀⠀⢀⡼⠃⠀⠀⠀⠀⠀⠀⠈⠙⢦⡹⣆⠈⢿⡿⣀⣀⣀⣀⠤⣄⠀⠀⠀⢸
⠀⡀⠂⠀⢀⡞⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣼⣧⡈⢧⢯⠀⠀⠀⠀⢸⡆⠀⠀⢸⢀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠲⢤⣀
⠀⠀⠀⢠⡟⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣼⢳⠘⣿⠀⠀⠀⠀⠀⡇⠀⠀⠸⢸⠀⠀⠀⠀⠀⢀⠈⠀⠀⠀⠀⠈⠓⠠⢤⡀
⠀⠀⢠⠟⠀⢀⠀⠡⡀⠀⠀⠀⠠⠤⢀⣀⡀⠲⢄⡿⠸⡄⣹⠀⠀⠀⠀⠀⣇⠀⠀⢀⢈⡀⠀⠀⢀⠔⠁⠀⠀⠀⢀⠄⠀⠀⠀⠀⠈⠲⢀
⠀⢠⡟⠀⢀⠈⠲⢄⠙⣄⠀⠀⠀⠀⢀⣀⣉⣛⠒⠚⢳⣁⣿⣄⠀⠀⠀⠀⢹⢰⣊⣀⣵⣮⠆⡀⠁⠖⠰⢧⢀⠔⠁⠀⠀⠀⠀⠀⢀⡴⠋
⣰⠋⠀⠀⠈⠳⢤⣀⠙⠪⢿⣟⠛⠋⠉⠉⢉⣉⡡⠤⠼⢷⣿⣍⡃⠀⠀⠀⢸⠶⠶⠖⠋⠁⣀⠜⠀⠀⢀⠔⠁⠀⠀⠀⠀⢁⠀⣰⠟⠒⠂⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠉⠙⠳⠶⠽⢷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠻⡧⠉⠀⠀⠀⡜⠀⠀⠄⠐⢮⠀⠀⠀⢀⠂⠀⠂⠀⠀⠀⡠⣣⣎⡙⠒⠒⠒⠐⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠻⣦⡀⠀⠀⠀⠀⠀⠀⢀⡟⠠⡤⠒⠚⠓⠒⠒⠒⠲⣦⣀⠀⠀⡞⠀⠀⠀⠀⡠⣪⡎⠙⢿⠷⠆⠀⠤
⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⢷⣄⠀⠀⠀⠀⠀⣛⠴⣿⠻⠀⣠⠆⠀⠀⠀⠀⠘⡎⢳⠔⠙⢷⡄⠠⣪⢴⣫⡞⢢⠈⠀⠰⣶⣴⡆⠤
⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠙⣦⡀⠀⠶⠛⠁⠸⠁⣠⠞⠁⠀⠀⠀⠀⠀⠀⣻⡄⡇⠀⠀⠙⠚⠁⠠⡝⠴⡆⡠⠊⡴⢠⣦⡌⢢
⠘⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⡴⠋⠙⢦⣄⠀⠠⠔⠋⠁⠀⠀⠀⠀⢀⣀⣀⣀⡝⠈⢁⡀⠤⠀⠐⠠⡀⠈⠳⣄⠑⠒⠠⠬⠍⠅⠼
⠀⣘⠤⢄⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠠⠊⠀⠀⠀⠀⠉⢻⡖⢶⡚⠛⠛⠉⠉⠉⢉⠉⠅⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠑⢄⠈⠁⠀⣀
⢰⠘⠧⢅⣒⠒⠂⠀⠀⠀⠀⢉⡉⣀⣄⣛⣚⠛⢲⣲⠤⣄⠙⢦⠙⢄⠀⠀⠣⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠄⠀⠈⠳⣄
⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠌⠃⠈⢳⡀⠣⡀⠀⠈
⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣿⣧⡾⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠋⠁⠀⠉⠻⣷⣤⠶⠂
⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠈⠉⠻⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⠀⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⢹⣿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡀⠀⠀⠀⠀⠀⢸⡿
⠀⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⢠⣿⠻⣷⣀⣀⣤⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣦⣀⣀⣀⣴⠟⠁
⠀⠀⠀⠀⣾⡿⣿⣄⠀⠀⠀⠀⢀⡼⠃⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡏⠉⢹⣿
⠀⠲⠶⠼⠋⠀⠀⣹⣿⡖⠒⢺⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⢿⣿⠀⠀⢸⣿
⠀⠀⠀⠀⠀⣠⡾⢻⣿⠀⠀⠀⣿⣏⠻⣦⡀⠀⠀⠀⠀⠀⠀⣿⣿⣶⣄⣴⣿⣿⡆⠀⠀⠀⠀⣠⡶⠋⠁⣾⡏⠀⢠⣾⢃⣴
⠀⠀⠀⠀⣴⠟⠀⣿⠇⠀⣀⠀⠸⣿⡀⠈⢻⣦⡀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿⡟⠀⢀⣠⡴⠟⠉⠀⠀⠠⠟⣁⣴⠟⠁⠘⣿⡆
⠀⠀⠀⣼⡏⠀⠸⢋⣴⠞⠁⢠⣤⣙⠃⠀⠀⠹⣿⡄⠀⠀⠀⠀⠘⢿⣿⣿⡿⠶⠞⠋⠁⠀⠀⠀⢀⣠⡴⠞⠋⠀⠀⠀⠀⢿⣇
⠀⠀⠰⣿⣀⣠⠶⠋⠁⠀⠀⠀⠈⠙⠻⠶⣤⣄⣽⠇⠀⠀⠀⠀⠀⠀⠙⢁⣀⣀⣀⣤⣤⠴⠶⠛⠉⠁⣶⡇⠀⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠈⠁⢠⣶⠀⠀⠀⠀⠀⠀⠀⢹⣶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⢠⣿⠃⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣼⡏⠀⠀⠀⠀⠀⠀⠀⠀⢿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣷⣤⣤⣤⣤⣤⣴⠿⠃
⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡏⠀⠀⠀⣿⡇
⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣷⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⣿⣇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣬⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠉⠉⠉⠉⢹⣿⠉⠉⠉⠉⢹⡿⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡿⠀⠀⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣧
⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⣸⡟⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⣰⡿⠀⠀⠀⠀⠀⢀⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠛⢷⣤⡀⠀⠀⠀⠈⠙⠛⠛⠖⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⠒⠛⠋⠁⠀⢀⣠⠼⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣠⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⣿⣿⣿⣿⣿⣿⠿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢿⣿⣿⣿⣿⣿⡟⠉⠁⠀⠀⠀⠀⠘⠿⠙⠿⠉⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⠃⠀⠀⠠⠶⣶⣿⡀⠘⢛⣒⡤⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣷⠀⠀⠀⣰⠋⠉⣩⡉⢻⡉⣤⡍⣦⣸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⢀⠁⠀⠀⠀⠻⣤⣀⣈⣤⠞⠳⣤⠴⠋⣼
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣤⣠⡁⠀⠀⠀⠈⠘⠉⠁⠀⠰⠶⠋⠀⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢧⡀⠀⠀⠀⠀⠀⠒⠛⠛⠛⠛⠓⣸⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⣆⠀⢀⡀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠃⠀⠈⠛⠷⠶⡶⣶⡞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣶⣶⣶⡄⠛⠁⠀⠀⠀⠀⠀⠀⠀⠘⢷⣦⣤⣄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣴⣿⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣀⣀⣀⣠⣴⣾⣿⣿⣿⣿⣿⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠈⠉⠛⠛⠻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⢠⡇⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⢻⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⠀⢸⣷
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠈⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡄
⠀⠀⠀⢄⠁⠀⠀⠐⠀⠀⠀⠀⡀⠠⠤⠄⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠐⠐⠀⠀⠀⠀⠀⢀⠞⠀⠀⢠⠠⡄⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⡀⠀⠀⠀⠀⡠⠄⠀⠀⠀⠀⢀⠀⠀⠀⠁⠀⣀⠀⠀⠰⠀⠀⠠⠄⢀⡀⠀⣀⠀⠴
⠀⠀⠁⠀⠈⠀⠉⠀⠀⠆⠀⠀⢀⡉⠉⢁⠀⠀⠀⠀⠀⠈⠀⠉⠀⠀⠀⠀⠀⣁⠈⢨⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠂⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⣀
⠀⠀⠀⠀⠐⠃⠀⠀⠀⠀⠀⠀⠒⠀⠀⠀⠂⠀⠀⠈⠁⠀⠀⠐⠂⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠀⠐⠂⠀⠐⠒⠀⠀⠀⠀⠀⠘⠘
⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠤⢀⠀⠀⠀⡀⠠⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠄⠀⠀⠀⠀⠀⠀⣀⢀⡀
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠙⠀⡀⠀⠰⠀⠀⠀⠸⠀⠀⠀⠠⠀⠀⠁⠀⠀⠀⠀⠀⠠⠀⠈⠁⠀⠀⠀⢀⠀⠁⠀⠀⠈⠀⡀
⠀⠀⠀⠆⢀⡄⠀⢠⡀⠲⠀⠀⠀⠠⠀⠔⠀⠈⠁⠀⠀⠠⠀⠠⠀⠀⠀⠀⠀⡀⠤⠠⠀⠀⠀⠀⠀⢀⡀⠀⠀⢐⠀⠀⠈⠀⠤⠀⠀⠔⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠠⠀⠁⠀⢤⠅⠀⠀⠈⠀⠀⠀⠀⠁⠀⠉⠀⠃⠈⠀⠀⠀⠈⠁⠃⠀⠀⢀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠁⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⠐⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⢉⠠⠀⠀⠀⠀⠀⠀⠈⠁⠉⠀⠀⠀⠀⠀⠁⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠒⠒⠂⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠐⠒⠂⠒⠀⠀⠀⠀⠀⠀⠚
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⢀⠀⠁⠀⠀⠘⠀⡀⠀⠀⢀⠀⠚⠀⠀⠀⣀⠀⠀⡀⠀⠀⠀⠔⠀⠀⠀⠈⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠃⠀⠢⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠤⠀⠠⠄⠀⠀⠀⠀⠂⠀⠠⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠁⠈⠁⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢄⠀⠀⠀⠀⠀⠀⢀⠄⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⢀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠀⠀⠘⠂⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢰⣾⣿⣧⠀⠀⠀⠂⠀⠀⠀⠀⠐⠀⠀⠀⢀⣠⣄
⠀⠀⠀⠠⣿⣿⣿⡆⠀⠀⢀⣤⣌⠛⢦⣬⣿⣿⣥⡤⠖⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⠇⢀⣠⠦
⠀⠀⠲⣄⣙⣿⣿⠏⠀⠀⠺⣿⡿⠃⠀⢨⣿⣿⠀⠀⠀⠀⢀⣴⣶⣦⠀⠀⠑⠲⢦⣭⣿⣿⡶⠋
⠀⠀⠀⠀⠉⢻⣿⣿⠆⢀⣾⣿⣿⣄⠀⣸⣿⣿⡇⠀⠀⠀⠀⣹⣿⣯⣤⠦⠀⠀⠀⢨⣿⣿⣄
⠀⠀⠀⠀⠀⣿⣿⣿⠀⣾⣿⣿⡏⠻⠆⣹⡿⣿⡇⠀⠀⠀⣰⡿⣿⣷⡀⠀⠀⠀⢀⣿⣿⣿⣿⡆
⠀⠀⠀⠀⢰⣿⠉⣿⠰⣿⠉⣿⡇⠀⠀⣿⣇⣾⡇⠀⠀⠀⠩⣾⠿⠿⢿⡆⠀⠀⠀⠈⡿⢸⡇
⠀⠀⠀⠐⣿⣿⢀⣿⠀⠀⠀⣽⡁⠀⠀⠿⠇⣿⡷⠀⠀⠀⠀⠙⠀⠀⠘⠁⠀⠀⠀⠀⣿⢸⡇
⠀⠀⠀⠀⣼⡇⠀⣿⠀⠀⠀⢸⠁⠀⠀⢠⡄⢸⡦⠀⠀⠀⠀⢀⠀⠀⢀⠀⠀⠀⠀⠀⡉⠘⡇
⠀⠀⠀⠀⠈⣿⣠⣿⠀⣯⣤⣿⡇⢀⠀⢿⡇⣹⡇⠀⠀⠀⢠⡡⡶⣶⠞⠁⠀⠀⠀⣀⣷⣸⣄⡀
⠀⠀⠀⠀⢀⢹⣿⣷⠄⠘⠿⣿⡷⠛⠀⢨⣻⣿⠇⠀⠀⠀⠀⢛⣿⡆⠠⠄⠀⠀⠀⠘⣿⣿⠋
⠀⠀⠀⠀⣡⣾⣿⡆⠀⠀⠰⠿⡷⠀⣀⣈⣿⣿⣀⣀⠀⠀⠀⠙⠛⠋⠀⠀⠀⠀⠈⢭⣿⣿⠓⠢⢀
⠀⠀⠀⠀⠉⠉⠉⠀⠀⠀⠀⠀⠐⠊⠀⠰⢾⠿⡂⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠶⠶⠶⠦⣤⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠲⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⢶⠞⠓⠒⠒⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠞⠉⠀⢰⣣⠴⠒⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠞⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡀⠀⡴⠀⠀⠀⠀⠀⠈⠉⠓⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣧⠘⠁⢰⣶⣄⠀⠰⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣄⠈⠻⠟⠀⠀⠙⠛⠃⠀⠀⠀⠀⠀⣀⣠⠄⢀⡼⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠙⠳⣤⣀⣀⣀⣠⣤⣤⠤⠴⠖⠚⠋⠉⣀⠴⠻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⣶⡈⠃⠀⠀⠀⠀⠉⠛⠲⠦⢤⣤⣤⣤⠤⠴⠖⠋⠁⠀⡠⠻⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠊⠀⠀⠹⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠉⠀⠀⠀⠀⠀⢻⡀⠙⣦
⠀⠀⠀⠀⠒⠲⠶⠤⠤⠤⢤⣤⣀⣠⠟⠳⢦⣤⣄⣀⠀⠀⠀⢀⣀⡤⠖⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧⠀⠈⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⣹⠄⢭⣽⣷⣞⣹⣶⡦⢤⣤⣄⣀⣀⣀⠀⠀⠀⠀⠀⠀⠸⡄⠀⠀⠙⣆
⠀⠀⠀⠀⠀⠀⢀⣀⡠⠤⠔⠒⠀⠙⠲⢶⠚⠁⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠛⠛⠚⠻⣿⠏⠛⠓⠒⠒⠶⢶⣠⠴⢚⣉⣀⣀⣀
⠀⠀⠀⠶⣎⣉⠉⠀⠤⣄⠀⠀⠈⠑⠠⠤⠀⠀⠀⠐⠤⢄⣀⠀⠀⠀⣠⠤⠤⠤⣴⣉⣁⣀⣠⡤⠴⠒⠋⠉⠀⠀⠀⠀⠀⠉⠉⠉⠉⠙⠛⠒⠒⠒⠶⠦⠤
⠀⠀⠀⠀⠀⠈⠙⠓⠶⠤⣍⣈⠁⠀⠀⠤⠄⠀⡉⠁⠒⠠⠀⠀⠉⢘⢧⣄⣀⣠⠤⠞⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠒⠶⠤⣄⣈⠙⠀⢀⡠⠔⠊⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠉

⠀⠀⠀⠐⠂⣐⣒⣒⣂⣐⣒⣂⣂⣒⡒⠒⢒⣒⣀⣀⣀⠀⣀⠀⠀⣀⣀⣀⣀⠀⣀⣀⢀⣀⡀⠀⠀⠀⠀⣀⣀⡀⢀
⠀⠀⠈⠁⠀⢻⣿⣿⢿⣿⣿⣿⣿⣿⡇⠀⢸⣿⣧⢿⣿⠸⣿⠿⠿⠛⠿⠏⣉⠀⣿⣿⡾⣿⠽⠿⣿⣗⣒⣿⡏⠁⢸⣶⠠⡤⠀⠀⣀⡠⠤⠤⣤⡖⢻
⠀⠀⣦⣀⠀⢸⣿⣿⠿⠶⠀⠐⣶⣶⡇⠀⢸⠉⢹⣿⣷⢲⣿⣷⣶⠒⣶⣾⣶⠀⠟⢻⣿⣿⣦⣤⣤⣤⠤⣀⣴⣦⢸⣽⣿⣿⠖⠋⠁⠀⠉⠀⠐⢿⣿⡀
⠀⠀⣿⣿⣿⣿⣿⣿⣶⣄⠀⣀⣿⢽⡇⠀⢸⠀⣸⣿⣿⣿⣿⠛⠛⠒⣛⣛⣻⠀⠀⢸⣿⣿⡾⠿⠿⠷⠤⠿⢿⣿⢸⣿⣿⣿⠀⢀⡀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠋⠉⠉⠉⠛⠉⠛⠛⠛⠛⠻⠿⠇⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠷⠿⠿⠿⠀⠿⠿⠿⠿⠲⠶⠶⠶⠰⠿⠿⠿⠘⠿⠿⠿⠿⠞⠃⠀⠀⠐⠒⠚⠛
⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣠⣤⡄⠀⠀⠀⣠⠀⠀⠀⠄⠀⣄⣤⣤⣤⣤⠀⣤⣤⡄⣤⣠⣤⠀⠀⠀⠠⢤⣤⣤
⠀⠀⣄⣾⡃⠀⠀⠀⠀⠀⢀⣤⣛⠋⠀⢉⡇⠀⠀⠈⣹⡿⠖⢓⡿⠻⢿⣿⣿⣿⣿⠀⣿⣿⠆⣿⣿⣯⣄⣀⠀⠀⢿⣿⣿⠷
⠀⠀⣾⣿⣷⡤⠶⣀⢀⣀⣾⣿⣿⣤⣀⣾⡇⠀⢸⣤⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⠀⣿⣿⡀⠛⢋⣿⣷⡯⠅⠐⢘⣉⣽⣶
⠀⠀⣿⣿⣿⡦⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⠀⣽⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣿⡟⠀⠚⣿⣿⣿⣷⣾⣿⣽⣿⣿
⠀⠀⠿⠿⠿⢷⠿⠿⠿⢿⣿⣿⣿⣿⣿⣿⡇⠀⠸⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠿⠀⣿⡟⠀⠀⢀⣿⣷⣷⣾⣿⣿⣿⣿⣿
⠀⠀⡀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⣠⣄⣀⠀⠶⠄⢰⡶⡖⠀⠀⠂⠐⠒⠒⠒⠲⠀⠀⢠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⡤⣤⢠
⠀⠀⡿⠦⠀⠀⣽⣿⣿⣿⡿⠿⠿⠿⠆⠀⠀⠀⠰⠿⡋⠁⣿⣿⣷⡆⢸⣿⣿⣷⣀⠀⠀⠀⠀⠀⠀⡄⠀⢸⣿⣿⣿⣿⣿⣷⣿⣿⣿⣿⣷⠃⢸
⠀⠀⣇⣠⣴⣶⣯⣭⣉⣉⣡⣤⣤⣀⡀⠐⡖⢻⣿⣿⣧⣴⡿⣿⣿⡇⢸⣿⣿⣿⣿⣧⠶⠄⢠⠔⢾⡇⠀⢸⣿⡇⠉⡻⢿⣿⣿⠿⠿⠿⠛⠀⢸
⠀⠀⣋⣹⣿⣿⣿⣿⣿⣿⣻⣿⣿⣿⡇⠀⣤⣿⣿⣧⣽⣿⣿⣿⣿⡇⢸⡿⣿⣿⣿⣷⡠⣤⠰⣾⢋⡇⠀⢸⣿⡇⢠⠉⢶⣼⣿⣦⣴⣦⠶⠿⢿⡀
⠀⠀⠀⠠⠿⠛⠛⠛⠛⠛⠷⠖⠒⠒⠂⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠘⠋⠙⠛⠛⠛⠛⠛⠒⠚⠛⠃⠀⠸⠿⠇⠸⠦⠼⠿⠿⠿⠿⠿⠄⠀⠸
⠀⢀⣀⡀⢀⡀⣀⣤⣤⣤⣤⣤⣤⠀⡤⠠⠄⣄⡀⢴⡶⠶⢂⣰⣶⠀⠀⣀⣀⣀⣠⣤⣄⡀⠀⠀⠀⠀⢤⠀⢰⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶
⠀⢸⣿⠷⣾⣿⣿⣿⣿⣿⣿⣿⣿⠀⣿⣾⣧⣬⣴⣶⣿⣿⣿⣿⣹⠀⣦⣍⣯⠉⠉⠉⢹⣿⡶⠒⣦⡤⣬⠀⢸⣿⣿⣿⣭⣽⣿⣿⣿⠟⠋⢀⣏⣈
⠀⠈⠉⣾⡄⢿⣿⣿⣿⣿⢿⣿⣿⠀⣿⣿⣿⣿⢫⣴⣿⣟⣛⢿⣿⠀⣿⣿⠉⣴⣶⣶⣾⣿⣤⣤⣿⣇⣿⠀⢸⡍⣏⠀⣿⣦⣄⣠⣤⣤⣴⣿⣿⣿
⠀⢀⠐⠒⣿⢿⡧⢴⠥⢶⣾⣋⣩⡄⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⡄⣿⣿⣤⣿⣿⣿⣿⣟⣿⣿⣿⣿⣿⠀⠘⣧⣿⣤⣿⣿⣿⣏⠉⠉⠉⠉⠉⣿
⠀⠈⠛⠙⠁⠐⠘⠛⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠛⠛⠛⠛⠛⠛⠃⠀⠀⠀⠀⠀⠛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⣶⣤⣤⣀⠀⠘⢿⣶⣄
⠀⠀⠀⠀⠀⢀⣤⣶⡾⠿⠛⠛⢛⣛⡛⢻⣿⣶⣶⣤⣍⣉⠛⠿⣷⣦⣌⠻⣷⡄⢰⣿⠂
⠀⠀⠀⣠⣾⠟⢋⣥⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⡿⢀⣀⣠⡄
⠀⠀⣴⡿⢋⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⠛⠁
⠀⠘⠛⢁⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣤⡀
⠀⢀⣼⡿⢋⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡙⢿⣧⡀
⢀⣾⠏⣰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⣻⣿⣿⣿⣿⣿⡿⠿⣿⣿⣿⠿⣿⣿⣿⡿⢿⣿⣿⣿⣿⣦⠹⣿
⣸⡟⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⢡⣾⣿⠟⢛⣭⣥⣴⣶⡿⠟⣿⣷⣿⡿⠟⠋⠀⠈⢻⣿⣿⣿⣿⡇⢻
⣿⡇⣾⣿⣿⣿⣿⣿⠏⣿⡿⠁⠀⠈⠉⠀⠀⠈⠉⠉⠉⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⢸
⣿⡇⣿⣿⣿⣿⣿⠃⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⢸
⢿⣇⣿⣿⡿⠻⠏⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣼
⠸⣿⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⣾⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣷⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⢀⣿⡟⠋⠁⠀⠀⠀⠀⠀⠀⠀⠙⠟⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠻⠋⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿
⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣿
⠈⢿⣧⣄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣾⠟
⠀⠀⠉⠛⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣀⣀⣀⣀⣠⣴⠗⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠟⠛⠁
⠀⠀⠀⠀⠀⠘⠿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠙⠛⠉⠉⠀⠀⠀⠀⠀⠀⠀⢀⣠⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠈⠛⠿⣶⣤⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣾⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠻⢿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣿⡿⠟⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⠟⠋⠉⠁⠀⠀⠀⠀⠈⠉⠛⠻⣷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣿⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠹⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢻⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣼⡟⠿⣶⣶⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣶⣤⣾⢿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⣼⡟⠀⢀⣿⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡄⠀⢻⣧
⠀⠀⠀⠀⠀⠀⣰⡿⠁⢀⣾⠿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⠻⣷⡀⠈⢿⣧
⠀⠀⠀⠀⠀⢰⣿⠁⠀⣼⡟⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⢻⣧⠀⠈⣿⣆
⠀⠀⠀⠀⠀⣿⡇⠀⣸⡿⠁⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⢿⣧⠀⠸⣿⡀
⠀⠀⠀⠀⣸⡿⠀⢠⣿⠇⠀⢸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣇⠀⠘⣿⡆⠀⢻⣇
⠀⠀⠀⢀⣿⠃⠀⣼⡟⠀⠀⣾⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⢹⣷⠀⠘⣿⡆
⠀⠀⠀⣸⡿⠀⢠⣿⠃⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠈⣿⡄⠀⢻⣧
⠀⠀⠀⣿⡇⠀⣸⡟⠀⠀⠀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⡀⠀⠀⢹⣧⠀⢸⣿
⠀⠀⠀⢿⣧⣠⣿⠇⠀⠀⠰⣿⣤⣤⣤⣤⣤⣤⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⡇⠀⠀⠘⣿⣤⣼⣿
⠀⠀⠀⠀⠉⠋⠁⠀⠀⠀⠀⢹⣿⠉⠉⠉⠉⢻⣿⠋⠛⣿⡟⠉⠉⠉⠉⣿⡏⠀⠀⠀⠀⠈⠉⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣼⣿⠀⠀⢹⣿⠀⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢻⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡏⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⠀⠀⢸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣄⣀⡀⠀⠀⣿⡇⠀⠀⢸⣿⠀⠀⢀⣀⣨⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣾⡿⠿⠿⠿⢷⣶⣿⠃⠀⠀⢈⣿⣶⡿⠿⠿⠿⢿⣷⣦⣀
⠀⠀⠀⠀⠀⠀⠀⢾⣿⣯⣤⣤⣤⣴⣶⣾⠿⠛⠁⠀⠀⠀⠛⠿⣶⣶⣤⣤⣤⣤⣽⣿⡇

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣠⣴⣶⠶⠛⠛⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡶⠶⠾⠛⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣌⠛⢦⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠻⣦⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠳⣶⣟⣻⣍⣷⡶⢶⡶⠶⠦⣤⡶⠶⠶⣄⡀⠀⠀⠈⠙⠷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣧⣿⣧⡘⠻⣾⠛⠛⠟⠻⠷⣤⡀⠈⢙⣦⠀⠀⠀⠀⠈⠙⢷⣤⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈⣷⢼⠇⠈⠛⢶⣄⡙⢶⣄⠀⠀⠀⠈⠛⠛⠁⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣮⣹⡇⠀⠀⠀⠀⠉⠻⢦⣌⠻⢦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⣿⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⠛⢿⡇⠀⠀⠀⠀⠀⠀⠀⠙⠳⣦⣉⠳⣦⡀⠀⠀⠀⠀⣀⣀⠠⠤⠤⣘⣛⣻⣤⡾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠿⠶⢾⣿⣦⡀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢷⣌⡙⠲⠖⠶⢂⣉⣭⡶⠶⠾⠋⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠞⢻⡏⠱⠀⢳⡜⠻⣯⠻⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠻⠟⢻⣿⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠉⠀⠀⢸⣗⠀⢠⣈⠇⠀⠙⠷⣮⡙⠶⢦⣀⣀⠀⠀⠀⠀⠀⣀⣀⣠⣾⣇
⠀⠀⠀⠀⠀⠀⠀⠀⣴⠟⠁⠀⠀⠀⠀⠀⠹⢧⣤⣴⣧⠀⠀⠀⠈⠙⠷⣦⣭⣤⣔⣒⢒⣷⣦⣬⣽⠟⠉⠹⣧⡄
⠀⠀⠀⠀⠀⠀⢠⡾⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠋⠉⠉⠀⠀⠀⠀⠀⠈⢿⡄
⠀⠀⠀⠀⠀⢰⡾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⡄
⠀⠀⠀⠀⢀⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⡀
⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠀⠀⢀⣀⣠⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⢸⡇
⠀⠀⠀⢠⡿⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠓⠈⣿
⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⢸⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣰⣿⣿⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⣷⡄⠀⢸⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⣿⣇⣸⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⣀⣿⡇⠀⢸⡇
⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠟⠉⠉⠻⢦⡄⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⠃⠀⢸⡇
⠀⠀⠀⠘⣧⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠁⠀⠀⠀⠀⠈⣷⡀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⢠⣿⠁
⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⠀⠀⠀⠀⠀⠀⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏
⠀⠀⠀⠀⠀⢻⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣤⣤⣤⣀⣀⣀⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠉⣿⠛⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠏
⠀⠀⠀⠀⠀⠀⠀⠘⠿⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠷⠶⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⢀⣰⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⠞⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠷⢦⣤⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠾⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣬⣿⡛⠷⣶⣤⣤⡀⠀⠀⠀⠀⠀⢠⣤⡶⠛⠛⣉⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠁⢈⡷⣦⡀⢀⡿⠃⠀⠀⠀⠀⠀⠘⣿⡀⣠⣾⡏⠉⣻⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡿⠃⢸⠀⠈⠛⢿⣷⠾⣷⣴⣶⣤⣤⣶⣼⡿⠋⠀⣅⠀⢿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢷⣤⡄⠀⠀⠀⠀⠈⠀⠈⠉⠀⠀⠀⠀⠀⢀⠀⠀⠈⣄⣼⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⢿⣶⡶⣄⣀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣷⣶⣾⠟⠁
⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣤⣤⣤⣤⡶⠛⠛⠉⠉⠀⠀⠉⠛⠛⠶⣤⣴⠶⠛⠛⠉⠀⠀⠀⠉⠛⠛⢶⣤⣤⣤⣤⣤⣀⡀
⠀⠀⠀⠀⠀⠀⣰⠟⠉⠀⠀⠀⢀⡛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⡀⠀⠀⠀⠉⠻⣆
⠀⠀⠀⠀⠀⣸⠏⠀⠀⠀⠀⠀⣼⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⠀⠀⠀⠀⠀⠻⣇
⠀⠀⠀⠀⣴⠟⠀⠀⠀⠀⠀⢀⡽⠀⠀⣀⣀⣀⣀⣀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⡀⠈⢿⡀⠀⠀⠀⠀⠀⠻⣆
⠀⠀⠀⣰⡟⠀⠀⠀⠀⠀⠀⠘⣷⡆⠀⢀⣀⣀⣤⣠⣤⠒⠂⣠⡀⠈⡁⢀⣄⡐⢲⣤⣤⣤⣄⣀⡀⠀⢰⣿⠃⠀⠀⠀⠀⠀⠀⠻⡀
⠀⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⢾⡿⠿⠿⠿⠿⠛⠛⠿⠿⠿⢿⣿⣿⣿⣿⠿⠿⠿⠿⠟⠛⠿⠿⠿⠿⢿⡷⠀⠀⠀⠀⠀⠀⠀⠀⢳⡄
⠀⢰⡟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢳⡄
⢠⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣷
⣾⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⣿⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠈⠛⠺⣶⡖⠿⠲⠷⠶⠾⠿⠶⠖⠒⠲⠿⠶⠶⠷⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠶⠾⠷⠶⠿⠖⠒⠾⠶⠶⠶⠶⠶⠶⠶⢲⣾⠖⠛
⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠋
⠀⠀⠀⠀⢿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡟
⠀⠀⠀⠀⠈⢿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡟
⠀⠀⠀⠀⠀⠀⠉⠛⠛⢻⡿⠛⠻⡿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠟⠛⣿⡟⠛⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢰⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⡄⠀⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⣹⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⡿⠦⠛⣷⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣾⠣⠴⢿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠻⣧⣤⣤⣤⣴⠿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠿⢂⣤⣤⣤⡾⠛
⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⡏⠁⠈⠉⠉⠉⠉⠉⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣦⣤⣤⣤⣤⣶⣤⣀⣹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠤⢀⡀
⠀⠀⠀⠀⠀⠀⢀⠠⠀⠂⠐⠀⢢⠊⠑⡄⠀⠀⠀⠀⠀⠀⢀⡤⠊⠁⠀⠀⠀⠀⠈⠃⣄
⠀⠀⠀⠀⡄⠒⠁⠀⠀⠀⠀⠀⠧⢤⢤⡙⠢⢄⠀⠀⠀⡠⠊⠑⠢⣄⠀⠀⠀⠀⠀⠀⠀⢣
⠀⠀⠀⠘⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⡰⠃⢀⠏⠀⠀⠀⠀⠀⠉⠢⢄⠀⠀⠀⠀⠀⠇
⠀⠀⡸⠉⢹⡀⠒⠂⠀⠀⠂⠐⠂⠇⠀⠈⠁⡀⠀⠀⠙⠤⡤⠄⠀⠄⠒⠂⠊⠉⡦⡀⠀⠀⡆
⠀⢘⠂⠀⠘⠄⠀⠀⠀⢀⠀⠀⢠⠀⠀⠀⢀⠋⠑⢄⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⡀⠈⣦⡜⢄
⠀⠈⠀⠽⠛⡀⠀⠀⠠⠀⠀⠀⠀⠒⢒⠶⠂⠀⠀⠈⡂⠐⢄⠀⠤⠄⠀⠀⠀⠀⠑⣉⠘⡄⠀⠱⡀
⠀⠀⠀⠀⠀⠈⢒⣤⣀⣀⣀⣤⡄⠚⠡⢠⠶⠀⡠⡘⠀⠀⢀⠡⣶⣤⢤⡤⢠⡶⢥⡒⢄⠘⠦⠴⠁
⠀⠀⠀⠀⠀⢠⠃⠀⡿⠀⡠⠒⢌⠫⡁⠐⢶⠋⠀⠀⠀⠀⠀⠀⡸⠀⠀⠀⠸⡀⠀⢱⢸⠦⢠⡄
⠀⠀⠀⠀⠀⠀⣴⠋⠀⠀⡆⠀⠀⠄⠈⢂⠀⢂⠀⠀⠀⠀⠠⣰⡇⠀⠀⠀⠀⠙⠎⠉⡈⢆⣤⠁
⠀⠀⠀⠀⠀⠈⠇⠀⠀⠀⠈⠢⢀⣀⠠⠞⠀⠀⠂⠀⠀⠀⠀⢹⠀⠀⠀⠀⠀⠠⠃⢀⠁⠀⠈
⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⣠⢇⣀⠐⠁⠀⠀⠀⠀⠈⠓⠒⠒⠒⠒⣄⣀⠄⠀⠐⠁
⠀⠀⠀⠀⠀⠀⠈⢁⡖⠒⠒⡔⢲⠛⡫⢴⡁⠂⠀⠀⠀⠀⠀⠀⠐⡴⠒⢄⢀⠝⠣⠤⡀
⠀⠀⠀⠀⠀⢀⣤⠪⣀⠀⣰⣁⠎⡰⠄⠀⠈⢆⠀⠔⠀⣀⣀⣀⢎⠀⠀⡘⢣⡔⠁⠀⠈⢱
⠀⠀⠀⢠⠊⠩⣀⡩⠂⠈⠁⠈⠧⠤⠊⠀⠀⡸⠀⠀⠐⠁⠀⠀⠁⠁⢸⠁⠈⠁⠐⠀⠀⡸⠀⢄⡀
⠀⠀⠀⠸⡀⠀⠀⠀⠀⡰⠀⠀⠈⠦⠄⠤⠐⠁⠀⠀⠰⢄⠀⠀⠀⠀⡌⠀⠀⠢⢀⡠⠞⠀⠀⠀⠐⠆
⠀⠀⠀⠀⠈⠈⠁⠂⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠈


⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣯⣽⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢩⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠭⠍⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⡸⠔⠒⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡠⠚⠁⠀⠀⠀⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠫⡀⠀⠀⠀⢀⡜⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⣠⠊⠘⡜⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⢹⠀⠀⢘⠜⡢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⠀⠸⡨⠬
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⡾⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡸⡙⠉
⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣈⣧⣘⣉⣾⣃⠀⢸⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢐⢸⠀⡰⡱⠁
⠀⠀⢰⣉⡉⢉⠉⠉⠉⠉⠉⢉⣉⣹⣹⠀⠀⠀⠀⡍⠀⢸⣸⠁⠀⠀⠈⣖⢖⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣒⣊⣸⣠⡱⠁
⠀⠀⢰⣟⣉⣉⣉⣉⣉⣉⣉⣋⣉⣉⡉⡆⠀⠀⢀⠃⠀⠀⢸⠀⠀⠀⠀⣀⠀⠀⠀⢀⣎⣀⣀⣀⣀⣀⣀⣀⣘⣀⣀⠀⠀⠀⠀⠀⢀⡠⠊⡥⢨⠑⢄
⢠⠤⠾⠥⠭⠬⠭⠭⠭⠭⠭⠬⠭⠽⠥⠦⠤⠤⠼⠤⠤⠤⠼⠤⠤⠤⠴⠯⠥⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠬⠤⠤⠤⠤⠤⠯⠥⠤⠵⠮⠤⠬⠽⠦
⠸⢤⡤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⠤⢤⡤
⠀⠀⠏⠉⠉⢉⠋⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⡉⠉⠉⠹
⠀⢸⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⡇
⠀⠸⠀⠀⠀⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⠀⠀⠀⠇
⠀⡄⠀⠀⢰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⢸
⢀⠃⠀⠀⡌⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠀⠀⠈
⢸⠀⠀⢀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡀
⡈⠀⠀⡸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇
⣇⣀⣀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣀⣀
⣿⣿⡿⠛⠿⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣷⣶⣶⣶⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⡰⠉⠉⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠙⡄⠡⢀⣀⠜⢀⠎⢢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠈⠐⠄⠀⠀⠘⢄⠈⠒⠤⠤⠔⠊⣠⠊
⠀⠀⠀⠀⠀⠀⠀⢀⠔⢣⠀⢆⡀⣀⠆⢠⠑⢄⠀⢱⠀⠀⠀⠀⢸⠀⠀⠀⠀⡔⠈⠐⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠑⠤⠀⠠⠔⠁⡠⠃⠀⢸⠀⠐⠂⠀⠘⢀⠔⢡⠀⢆⡀⡀⠎⢠⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⡎⠀⠀⠀⢸⠀⡆⠀⠀⢨⠀⠣⡀⠑⠤⠀⠠⠔⠁⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠄⢄⠀⡇⠀⠀⠀⠘⠤⠃⠀⠤⠆⠀⠀⠈⡆⠀⠀⠀⠀⡎
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢸⠀⠨⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⢀⠄⠄⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠼⠀⠸⠤⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠸⠀⠸⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠧⠤⠀⠠⠤
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣉⣉⣉⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡠⠚⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⠋⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡀⠀⢳⡄⠀⠀⢀⣠⡶⠟⠋⠀⠀⠀⠀⣴
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠳⢤⡈⠉⠳⣾⠛⠉⠀⢠⡤⠶⠤⢤⣴⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠁⠙⠷⡀⠀⠈⠳⣄⠀⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡄⠀⣸⠃⠀⠀⠀⠈⡠⠀⠀⠈⠻⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣀⢀⡟⠀⠀⠠⠀⡐⠀⠀⠀⠀⡀⣽⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣌⠻⣔⣾⠐⠀⠀⠄⠀⠀⠀⠀⢰⣿⠘⠀⠀⠀⠀⠀⠀⣦⠘⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢰⢡⠻⣿⠟⢛⠛⠿⢶⡆⠀⠀⠀⡾⠉⠐⠶⡀⠀⢀⡀⠀⠉⠉⠉⠀⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⡾⠚⠉⢖⡉⠡⣠⡜⡇⠀⠀⣸⣧⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠈⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠐⠀⢤⠈⡆⢈⡾⠁⠀⠰⠿⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠐⠀⠀⠀⣁⡅⢸⡇⠀⢰⣶⠀⠀⠳⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠤⠾⠾⡄⠀⠀⠴⠐⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⢦⣤⣭⡵⠿⠅⢀⣼⣟⠀⠀⠀⢳⡄⡀⠀⠀⢀⣀⡤⠒⠉⠀⠀⠀⠀⠀⠘⠀⡠⢤⡅⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠀⡀⠻⢯⡁⠀⠀⠘⠛⠁⠀⢸⠇⠀⢹⠟⠒⠚⠛⠿⠆⠀⠀⠀⠀⠀⠀⠀⠀⢠⠣⣼⣇⠀⠀⢀⡤
⠀⠀⠀⠀⠀⠐⣶⠀⢀⣴⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣆⣠⢾⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣜⣇⢧⢼⠶⠖⠋
⠀⠀⠀⠀⣰⡖⠉⠁⠉⠹⠀⠀⠀⠀⠀⠀⠀⠐⠒⠀⠀⠀⠀⢸⠌⣧⣳⡾⡆⠀⠀⠀⠀⣰⡀⣀⡄⠀⠀⠀⠿⢿⡋⠉⡍
⠀⠀⠀⠀⢸⡇⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣶⣿⣿⣷⣇⠀⠀⠀⠐⠋⠉⠋⠀⠀⠀⠀⠀⠀⠃⢠⠀⠀⢸⠃
⠀⠀⠀⢀⣟⡇⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⣄⣼⣳⣴⠚⠉⠡⢼⣿⣿⣿⢽⣶⣆⡀⠀⠀⠀⠀⠀⠀⠀⡀⣄⠀⠀⢸⠀⠀⠿
⠀⠀⠀⠀⠉⠛⠛⠋⠉⠙⠒⠶⠶⠶⠖⠒⠒⠋⣽⣿⡓⠒⠒⣀⣈⣉⣉⣟⠾⠋⠀⠉⠓⠢⢀⢀⣤⣤⠴⠖⠛⢲⠦⢼⠒⠚⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⠀⠀⠦⣄⡀⠀⠀⠉⠉⢉⣠⠼⠛⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠀⠀⠀⠉⠒⠒⠋⠉⠢⡀⣠⠶⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣞⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡱⡒⢼⣿⢐⢱⣶⢶⣲⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣮⣑⠹⡏⠞⡁⣨⣏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⢄⡀⠀⢀⠶⣉⡳⠒⣚⡙⠐⠂⣛⠩⣿⠷⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⢱⣳⣄⡈⢑⠔⣆⠙⢿⢋⢓⢂⣾⢰⡤⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⣯⡨⡻⣏⢦⠟⡩⢍⣩⠟⣽⡇⢸⡷⢄⣯⣑⠲⠤⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣙⠉⠩⡙⠧⢓⡛⣫⠿⠿⠿⣤⡉⠛⠾⢿⣖⡈⠉⠛⠿⣷⣦⣭⣳⠶⠤⣀
⠀⠀⢀⣀⣀⡒⠒⡠⠔⠲⠖⠒⠙⡿⢛⠭⣠⠄⡄⡻⣿⣷⣿⡾⠛⣢⣦⣀⠈⠙⠷⣢⣄⡀⠉⠻⣧⣾⣿⣎⡦
⠀⠤⠤⠤⠤⠀⢸⣰⢿⣷⢀⣤⣞⣠⣵⣾⢱⢡⣿⣍⢪⣿⣯⣶⣾⣿⣿⣮⡑⠤⡀⠈⠑⠫⣖⣶⡮⣉⣭⡳⣯⠭⠤
⠀⠀⠛⠋⠁⣀⣘⣼⣜⣋⣰⣿⣿⣿⣿⣿⣂⣾⣛⡛⠓⠬⠿⢛⣛⣛⣿⣿⣿⡗⢰⣿⣿⡷⣽⣿⠸⣿⢿⢟⡴⠒⠒⠂
⠀⠀⠀⠀⠀⠀⠰⠶⠶⠖⠂⠀⠀⠀⠀⠀⠺⠷⠶⠶⠤⠤⠤⠤⠭⠭⠭⠭⠽⠷⠾⠷⠿⠿⠿⠿⠷⠶⠯⠭⠭⠭⠅
⠀⠀⠀⠀⠀⠀⠉⠉⠉⠛⠛⠛⣛⣛⣛⣛⠛⠛⠛⠛⠓⠒⢀⣀⣀⣀⣀⣐⣒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣶⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⢿⣿⣦⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⣿⣿⣿⣿⣿⡿⡿⠛⣷⣿⣷⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡄⠀⢹⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣷⠀⢸⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇⢸⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⢛⣛⣙⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢛⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⠉⠉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⣉⡉⠉⢹⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡟⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠙⣿⠀⢸⡇⠀⠀⠀⠀⠀⢀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⢠⡶⠛⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢀⣴⠞⠁⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠸⣧⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⣤⠟⠁⠀⣠⡾⠃⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠉⢳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇⠀⠀⠀⠀⠀⠀⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⣧⣀⣀⡀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣿⠀⢸⡇⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠉⠉⠉⠁⠀⠀⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁⠀⢸⡇⠀⠀⣿⠉⠉⠉⠉⠉⢹⡏⠉⠙⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢶⣶⣴⣤⣶⣶⢶⣴⣦⣶⣶⣶⣶⣶⢶⣶⢶⣶⣶⣶⣶⡶⠾⠃⠀⠀⣿⠀⠀⠀⠀⠀⢸⣧⣤⣴⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣠⣿⣀⣀⣀⣀⣀⣀⣠⣿⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⡀⠀⠀⠀⠀⢸⡏
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⣤⠀⠀⢠⣤⣤⣿⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣭⣽⣧⣤⣤⣤⣤⣤⣤⣤⣼⣷⣤⣤⣤⣴⣿⣥⣤⣤⣤⣤⣤⡄
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢼⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠛⢻⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣧⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠶⠶⠶⠆⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⡟⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⢸⣇⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣤⣼⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠀⠀⠀⠉⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈


⠀⠀⢀⠀⠀⢀⠀⡀
⣿⡟⣉⣻⠟⠿⠿⠿⠿⠟⠿⠿⠿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣷⣦⣄⠀⠀⣠⣤⣤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠛⠉⠛⣷⣄⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣆⠀⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⠋⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⢸⡟⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠟⠁⠀⠀⠀⠀⠀⠀⢀⣹⣷⣄⡀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⢀⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡿⠃⠀⠀⠀⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣧⡙⢿⣿⣿⣿⣿⣿⣿⠟⠁⣼⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⠏⠀⠀⠀⠀⢀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣧⠀⠉⠛⠛⠛⠋⠁⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣛⡁⠀⠀⠀⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⢷⣤⡀⠀⠀⠀⣠⣾⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣾⣿⣿⡆⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋⠀⠀⠉⠛⠿⠿⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠿⠟⣵⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⡁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⢿⣿⣿⣿⡿⠟⠉⠈⠙⠻⢷⣦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠻⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⠿⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⠟⠋⠁
⠀⠀⠀⠀⠀⠀⠀⣀⣴⠿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡾⠛⠉
⠀⠀⠀⣀⣤⣶⠿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⡿⠛⠁
⣴⣾⣿⡛⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⣿⣷⣶⣤⡀
⢿⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⢿⣿⣿⣿⡿⠇
⠈⠻⣿⣿⠃

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣄⠲⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠘⡆⣷⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⡗⢺⠀⢛⡃
⠀⠀⠀⣤⣤⣤⣤⣤⡄⠀⠅⢠⠇⠘⠀⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⣠⣤⡀
⠀⠀⠀⢃⣀⣀⣀⣀⡁⠀⠀⠈⠀⠀⡴⢿⣶⠀⣏⠹⡀⠠⠀⠀⠀⠀⡟⢹⡀⠀⠀⠀⠀⠀⢇⠰⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡙⢸⡿⢰⠲⠶⠏⢳⡀⠀⠀⢀⣵⡜⢿⡀⠀⠀⠀⣠⠒⠴⠀⣆
⠀⠀⠀⣠⠤⠄⠀⠀⠐⡶⠆⠀⠀⢸⠃⠈⢁⣿⠀⠀⣦⢸⡇⠀⢰⠫⣇⣀⡀⣧⠀⠀⢀⡏⣶⠀⡀⢹
⠀⠀⡀⣀⡉⣉⡀⠀⠀⣒⣋⠂⠀⣼⡇⠀⢸⣏⢀⣀⡿⢀⡇⠀⠈⠈⡡⠤⣁⠏⠀⠀⠈⡇⢸⣐⠁⣸
⠀⠀⠰⠾⠶⠶⠶⠶⠶⠶⠦⠀⠀⡏⣧⡄⢸⢿⣿⣻⡇⣸⠀⠀⢸⠆⢀⠀⢸⠀⠀⠀⠀⣷⡘⣠⢠⡇
⠀⠀⢀⡴⠒⣂⠐⠲⠀⠀⣠⢄⠀⢷⣿⡇⠈⢹⣿⣿⣿⢸⠀⠀⢸⠀⣾⡆⢸⡀⠀⠀⠀⢻⡞⠹⡄⠁
⠀⠰⣬⣀⣀⣥⣼⠄⠀⣸⣤⣌⣀⢸⣿⡀⠀⣿⣿⣿⣿⣾⠀⠀⢸⠀⡏⣧⢸⡇⠀⢰⢶⣞⡟⣟⢉⠉⡏
⠀⢧⢨⠉⢩⠉⠀⠀⢠⣧⣸⠩⠀⡆⣿⡇⠀⢻⣿⣿⣿⣿⠀⠀⢸⠀⡇⢻⢸⡇⠀⠈⣿⣿⣷⡿⣬⡄⠁
⠀⠀⢠⣄⣀⡀⠀⠀⠈⠨⠠⠤⡤⠀⢘⣷⠀⢸⣟⣹⣿⡄⠀⠀⠈⡀⠃⠀⢘⣇⡀⠀⠀⠀⣿⠁⠹⡿⠤
⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠛⠛⠋⠁⠀⠀⠀⠉⠉⠉⠉⠈⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠈⠀⠂⠀⠂⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠤⠀⢢⣀⠀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⠀⠀⡠⠂⠁⢀⠔⠀⠀⠀⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⡄⠀⠀⠀⠤⠀⠈⠀⠀⡰⡄⠀⠀⠀⢀⠊⠀⣀⠖⣠⡄⠐⠈⠤⢂⠠⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡂⡄⠁⠀⠁⡹⠏⠀⠀⠀⠀⠲⠀⠀⠀⠀⠀⠉⠀⠀⠘⢿⠁⠀⠢⠔⠈⠀⠀⠀⠀⠠⠡⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠐⠐⠀⠀⢠⣄⠀⠀⠀⢰⣦⠀⡀⠀⠀⠘⠀⡠⠋⠀⢀⠀⠀⢀⣅⠀⠀⠀⢾⡄⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣀⠀⠀⠀⠀⠛⠀⠀⠐⡌⠁⠀⠁⠀⠀⠀⠀⡇⠀⡆⠄⠀⠀⠀⠛⠁⠀⠂⡌⠀⠀⣀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠄⠀⠀⠀⠀⠀⢀⣠⠄⠁⢀⠀⠀⠀⠀⠀⢡⢣⠀⣸⡄⠀⠀⠀⠀⠀⠄⠄⠀⢀⢜⠎⡘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠐⠀⠀⡀⠀⠀⠀⠠⠀⠁⠀⠀⠀⠀⠀⠀⠡⢳⣇⡈⠀⢄⡀⠀⢀⣀⡠⠔⠃⠘⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠆⢂⢸⣟⠦⠤⠵⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡃⠀⠆⠈⠀⢀⠉⠀⣠⡄⡄⠈
⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⢌⠢⡑⠜⠘⠚⠒⠂⠀⠀⠶⠂⠠⠀⠲⠄⠀⠀⠀⠒⠒⢑⠁⠀⡀⠀⠀⠀⠀⠀⠀⠉⠈⠢⠴⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠢⡝⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠒⠒⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⢐⠠⠽⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⡒⠒⠲⠒⠲⠶⠶⠤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠉⠀⠀⢺⠀⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣌⠀⠺⠂⣠⡉⠰⠦⠈⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⡁⠐⠗⢀⡉⠀⡦⠀⠘⠃⢠⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠒⠤⠜⢁⠤⠄⠸⠟⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄⠀⠀⠀⡐⠀⠀⠀⢀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠐⠀⠀⠀⠀⠈⠀⠐
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠨⣀⠀⠀⡆⠀⠀⠀⠀⠸⠀⠀⢀⣡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⡐⠁⠀⠀⠀⠀⠀⠄⠠⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣅⡈⠉⠡⠃⠀⠀⠀⠀⠀⠏⠉⠁⢈⡤⠆⠀⠀⠀⠀⠀⠀⢌⡁⠀⠀⠀⠀⠀⠀⠀⢈⣁⣀⠥⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠓⠚⠁⠀⠀⠀⠀⠈⠉⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣠⠤⠔⠒⠚⠉⠁⠀⠀⠑⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠴⠒⠒⠉⠁⠀⠀⠀⣀⣀⢤⣴⣶⡄⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⠤⠔⠒⠊⠉⠀⠀⠀⣀⣀⣤⣴⡴⠶⣿⡏⠻⣷⣦⡻⠿⠛⠒⠀⠀⠀⠀⢈⣲⡀
⠀⠀⠀⠀⠀⠀⢴⡉⠀⠀⠀⠀⠠⣤⣶⣦⡙⣷⣝⣿⡄⠻⡷⠀⠈⠛⠀⠈⠀⠀⢀⣀⡠⠤⠒⠒⠉⠁⠀⠁
⠀⠀⠀⠀⠀⠀⢸⠗⢄⠀⠀⠀⠀⠙⢿⡿⠿⠋⠉⠉⠁⠀⠀⢀⣀⡠⠤⠒⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⢸⠘⡄⠑⣄⠀⠀⠀⠀⠀⠀⣀⡠⠤⠖⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣀⠀⠀⠀⢸

⠀⠀⠀⠀⠀⠀⢸⠀⠱⡀⠈⣱⣤⠤⠖⠋⠉⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣤⣴⢴⣿⣋⣿⣿⣏⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⢸⠀⡀⠋⠉⠀⡇⠀⠀⠀⠀⠀⠀⠀⣠⡄⣶⠾⣿⠋⠉⣿⠁⠿⠟⠈⠋⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⢸⠀⡇⠀⠀⠀⡇⠀⠀⠀⢰⣿⣿⣇⣿⣧⡟⠀⠟⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⠃
⠀⠀⠀⠀⠀⠀⠈⢆⡇⠀⠀⠀⡇⠀⠀⠀⠸⠿⠟⠃⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠒⠊⠉
⠀⠀⠀⠀⠀⠀⠀⠈⠣⡀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⠔⠂⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢄⠀⡇⠀⠀⠀⠀⢀⣀⠤⠄⠒⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣧⠤⠔⠂⠉⠁

⠀⠀⠀⠀⣀⣷⣴⣶⣦⡄⠀⠀⠀⣠⠤⠤⠤⠤⠤⡀⠀⠀⢀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⡀
⠀⠀⠀⣾⠟⠿⠟⢛⡟⠁⠀⠀⠀⡇⣷⠛⠛⣛⠃⡇⠀⠀⠀⠀⢠⣬⣿⢗⡀⠀⠀⠀⢤⠀⠀⠀⠸⠯⣿
⠀⠀⠀⢸⡃⠰⠆⢚⠆⠀⠀⠀⠀⢧⣿⣤⣤⣬⣤⠇⠀⠀⠀⠀⠘⢛⣴⣾⣿⡆⠀⠀⠀⠱⣆⠀⠀⠀⢠⠆
⠀⠀⠀⠀⠣⣀⡠⠎⠀⠀⠀⠀⣴⣿⣿⣯⣽⣿⣿⣷⠄⠀⠀⠀⠀⠀⠙⠿⠿⠃⠀⠀⠀⠀⠈⢳⠀⠀⠁

⠀⠀⢰⠶⠶⠶⠶⠶⠶⡦⠀⠀⢰⢒⠒⡒⠒⠒⠒⣦⠀⠀⠀⢠⢶⠶⠶⡶⡄⠀⠀⠀⠀⠀⠒⠒⠒⢢
⠀⠀⠀⢃⠀⠀⠀⠀⠰⠁⠀⠀⢸⠀⠀⢸⣖⣖⡦⣿⠀⠀⠀⢸⡃⠄⠤⠀⡇⠀⠀⠀⣠⣶⣤⣤⣤⣼⣦⡀
⠀⠀⢀⠔⠛⠛⠛⠛⠣⡀⠀⠀⢸⣬⣭⣭⣤⣬⣤⡿⠀⠀⠀⢸⣣⣀⣰⣆⡇⠀⠀⠀⣯⡟⠉⠉⠉⠉⣿⡇
⠀⠀⠘⠻⠛⠛⠛⠛⠛⠋⠀⠀⠀⠀⠀⠾⠿⠄⠀⠀⠀⠀⠀⠸⠿⠿⠿⠿⠃⠀⠀⠀⠀⠡⠤⠄⠤⠤⠃
⠀⠀⠀⠀⠀⢀
⠀⠀⠀⣰⣾⣿⣿⣶⣆⠀⠀⠀⠀⢶⡀⠀⠀⠀⠀⠀⠀⠀⠀⣰⠀⢸⡆⠀⡆⠀⠀⠀⢰⠋⠉⠉⠉⠉⡆
⠀⠀⠀⣿⠟⣋⣁⡙⣿⠀⠀⠀⠀⢀⣽⣶⢧⡆⠀⠀⠀⠀⠀⣿⠀⢸⡇⢸⡇⠀⠀⠀⢸⠀⠀⠀⢶⣄⠇
⠀⠀⠀⢹⣦⠁⡄⣡⡿⠀⠀⠀⢰⡷⠾⡿⠿⣏⡽⠃⠀⠀⠀⢿⠀⢸⠃⠘⠗⠀⠀⠀⢸⡄⠀⠀⠀⠙⢷⣄
⠀⠀⠀⠀⠻⠉⠊⠹⠁⠀⠀⠀⠀⠙⠚⠛⠋⠉⠂⠀⠀⠀⠀⠻⠀⠠⠀⠈⠟⠀⠀⠀⠘⠧⠤⠤⠤⠤⠏⠙

⠀⠀⣠⣤⣄⠀⢀⠀⡀⠀⠀⠀⢠⢲⡀⡔⡆⡖⠶⡄⠀⠀⠀⠀⢀⡖⢲⠀⠀⠀⠀⠀⠀⠀⠀⣂⣀
⠀⠀⣿⣿⡿⣿⣿⣿⣿⢸⠀⠀⠀⠉⣷⠉⠀⡇⢸⠁⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⡎⠀⠈⡃
⠀⠀⣿⣟⣤⣹⣉⣸⣿⢸⠀⠀⠀⠀⣾⠀⠀⡇⢸⡀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠹⣀⣰⠁
⠀⠀⠹⠿⠿⠿⠿⠿⠿⠞⠀⠀⠀⠀⠈⠀⠀⠧⠼⠀⠀⠀⠻⠿⠿⠿⠿⠿⠿⠟⠀⠀⠀⠀⠀⠻⠋

⠀⠀⠀⠐⣧⡀⣤⣤⠀⠀⠀⠀⠀⠀⠀⣴⣶⡄⠀⠀⠀⠀⠀⠀⢠⣶⡆⠀⠀⠀⠀⠀⠀⢀⣴⣄⢀⣴⠆
⠀⠀⠀⠀⠈⢷⣝⣋⡀⠀⠀⠀⠀⠀⢀⣬⣭⣄⠀⠀⠀⠀⠀⢀⣠⣭⣤⡀⠀⠀⠀⠀⠀⢀⣭⣥⡾⠃
⠀⠀⠀⠀⠀⢸⣿⣿⡿⣦⠀⠀⠀⣴⢿⣿⣿⣿⢷⡄⠀⠀⢰⣟⣿⣿⣿⣻⡦⠀⠀⢀⣴⣿⣿⣿⡇
⠀⠀⠀⠀⠀⣿⣿⣿⣿⠋⠀⠀⠘⠁⣼⣿⣿⣿⡄⠛⠀⠀⠀⢹⣿⣿⣿⡏⠀⠀⠀⠛⢁⣿⣿⣿⣇
⠀⠀⠀⠀⠀⢹⡏⢻⡏⠀⠀⠀⠀⠀⠉⣿⢹⡏⠀⠀⠀⠀⠀⠈⣿⠉⣿⠁⠀⠀⠀⠀⠈⢹⡏⣿⠋
⠀⠀⠀⠀⠀⢸⡇⢸⡇⠀⠀⠀⠀⠀⠀⣿⢸⡇⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢸⡇⣿
⠀⠀⠀⠀⠀⠘⠃⠘⠁⠀⠀⠀⠀⠀⠀⠉⠈⠁⠀⠀⠀⠀⠀⠀⠉⠀⠉⠀⠀⠀⠀⠀⠀⠘⠃⠙


⠀⠀⠀⠀⠀⠀⠀⠸⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⣷⡀⣰⣶⡀⣰⡗
⠀⠀⠀⠐⠶⠶⠶⣶⣶⣶⠶⠶⠶⠂⠀⠀⠀⠀⠻⣦⠠⣾⡆⢠⡿⠁⠀⠀⠀⠀⠀⠈⢷⣬⣭⣼⠏
⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⡀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣷⣶⣶⠟⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣷
⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⢠⣤⣤⣴⣿⣿⣿⣧⣤⣤⣤⠀⠀⠀⠀⣠⡟⠉⠉⠹⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣿⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠼⠋⠀⠀⠀⠀⠈⠿



⣿⣿⣿⣿⣻⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣟⣿⣿⣻⣛

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠠⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣟⠀⢈⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⢀⣴⣾⣿⣮⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠋⠁⠈⠱⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⣾⢏⣿⡿⠻⣷⣤⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⢁⡼⠀⢠⣄⠉⠻⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠿⣶⡆⠉⠀⢰⠶⠶⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣈⠉⢻⣾⣿⡏⣋⣹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣿⠀⣿⡃⡶⠶⠞⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⡇⠀⣿⠀⣾⢩⣭⣥⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⢸⣟⢠⡟⠓⠃⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣭⣭⡅⠸⡏⣭⣯⣼⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⡾⠛⠚⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⢩⣤⣥⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠒⠒⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣭⣭⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⢀⣠⠄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⢠⠊⠀⠀⠀⡈⠂⠀⠀⠀⠀⠀⠀⡒⢤⠀⢠⠈⢁⠈⠁⠑⡄⠀⠀⠀⠀⠀⠀⠀⠔⠉⢀⠀⠀⢃⠀⠀⠀⠀⠀⠀⡴⣦⢭⣴⠎⢣
⠀⠀⠀⠀⠈⠢⠁⢂⡊⠔⠁⠀⠀⣀⠀⠀⠀⠈⠢⢀⠈⠢⣀⠀⠀⡀⠃⢀⣀⢄⠀⠀⠀⠀⡖⣔⣁⣀⠠⠊⠀⠀⠀⠀⠀⠀⠢⣘⠶⢊⡠⠌⣼
⠀⢀⣀⡄⠀⠀⡠⠈⠃⠂⠀⠀⠐⠈⠙⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠤⠤⠔⠀⠀⠀⠀⠀⢠⠓⠩⠈⠉⠐⠂⠀⠀⠀⠀⠀⠀⢀⠔⢙⠣⠄⠠⠂
⠀⠈⠀⠘⠀⠁⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⢠⡇⡒⠁⠀⠀⠀⠀⠀⠀⠀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠊⠀⠀⠀⠀⠀⠀⠀⠀⠠⠅⢺
⠀⠀⠀⠀⠀⠀⠀⠆⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠂⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⠂
⠀⠀⠀⠀⠀⢀⡎⢄⠀⣇⠀⡀⠀⠀⠀⠀⠀⠀⢀⠄⣀⡄⠀⠈⣄⠀⣀⠀⠀⠀⠀⠀⢀⠀⣠⠎⠀⠘⣄⡀⢄⠀⠀⠀⢀⠀⣠⠄⠀⠈⣄⡀⢀
⠀⠀⠀⠀⠀⠈⠉⠉⠀⠉⠉⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢠⠀⠀⠀⠠⡐⠄⠀⠀⠀⠀⠀⠀⠀⠀⡀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⣀⠀⢀⠀⠀⡍⠆⠀⠀⠀⠀⠀⠀⠔⠂⠉⠀⠁⠄⠀⠀⠀⠀⠀⡰⠀⢃⠄⠒⠀⠒⢏⠘⡄⠀⠀⠀⠀⢠⠂⡉⠀⢀⠈⡆
⠀⠀⠀⠀⠀⠰⠊⠁⣀⠠⠁⠀⠈⠀⠀⠀⠀⠀⠀⠘⢄⠀⠈⠉⡀⠆⠀⠀⠀⠀⠀⢃⠀⡁⠐⠀⠁⠂⡠⠀⠄⠀⠀⠀⠀⠐⠄⣉⣀⡀⠔⠁
⠀⠀⠀⠀⠀⠢⣀⠀⣀⡠⢊⠠⠊⠀⠀⠀⠀⠀⠀⠀⠀⡨⠻⡅⠀⠀⠀⠀⠀⠀⠀⠀⠑⠈⠐⢲⠂⠈⠐⠀⠀⠀⠀⠀⠀⠀⠀⠀⡿⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⡠⠒⠈⠉⢃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠁⠤⣀⢤⠀⠀⠀⠀⠀⠀⠀⠀⢘⠀⠀⠀⠀⠀⠀⢀⠰⢢⡀⠠⠊⠸⠈⠐⠀⠂⠌⠘
⠀⠀⢠⡀⡚⠀⠀⠀⠀⢸⢀⢀⠀⠀⠀⠀⠀⠀⠀⠀⡞⡦⢰⠀⠀⠈⠂⠙⠀⠀⠀⠀⠀⠀⠀⡸⡀⠀⠀⠀⠀⠀⠈⠒⠈⠀⠀⠀⡆
⠀⠀⠀⡠⣄⠀⠀⠀⡠⠊⠀⠀⠀⣔⠬⠓⠀⠀⠀⠀⠑⡠⠂⠈⢂⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇
⠀⠀⠃⠈⠀⠀⠀⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⢉⠀⠀⠀⢘⢀⠀⠀⠀⠀⠀⠀⢀⡀⡐⠀⠀⣀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢁⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁⠀⠀⠉⠐⠘⠀⠀⠀⠀⠘⠁⠊⠁⠀⠀⠁⠀⠃⠀⠀⠀⠀⠀⠀⠀⠈⠁⠘⠛⠃
⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⡠⠐⠈⠁⠈⠠⠀⠀⠀⡠⠐⠈⠀⠈⠢⠀⠀⠀⠀⠀⡠⠀⠘⠃⠐⢄⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠁⠐⠢⢤⠀⠀⠀⢀⠴⠖⠛⠙⠳⣶⢦⠤⠂
⠀⠀⢃⠀⠀⠐⠊⡠⠁⠀⠀⢀⠀⠄⠀⢀⠰⠀⠀⠀⠀⠀⢄⠀⠀⠠⠀⡸⠀⡠⡆⠀⠀⠀⠀⠀⠀⢄⠀⠐⢄⠀⡆⠀⠀⠨⡀⠀⠠⠤⢀⠿⢦⠈⠍
⠀⠀⠀⢈⠜⣧⡁⠀⠀⠀⠀⠀⠉⢀⠞⠣⡀⠀⠀⠀⡗⠠⣀⣁⠰⡖⣈⠤⠊⢀⠧⠀⠀⠀⠀⠀⡠⠞⠳⠠⠤⠔⠀⠀⠀⠀⠈⠐⢲⠆⠥⠀⠊
⠀⠀⠀⠂⠀⠄⠈⠂⠰⠭⠑⠀⠒⠁⠇⠀⢠⠀⠀⠀⡍⡀⠀⠁⠒⠖⠁⠀⠀⠘⠆⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠌
⠀⠀⠰⢢⠀⡁⠀⠀⠀⠀⠀⠀⠀⠀⢠⠰⠉⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⣄⢀⡄⠀⡀⠀⠀⠀⠀⠀⢀⢀⠌⢀⡀
⠀⠀⠐⠄⠀⡧⠀⠀⠀⠀⠀⠀⠀⠀⡟⠈⠋⠀⠀⠀⠀⠀⠁⠂⢤⠼⡄⠐⠈⠀⠀⠀⠀⠀⢈⡜⢃⠡⠪⠱⠀⠀⠀⠀⠀⠬⠄⠀⢸⠇
⠀⠀⠀⠀⠀⠇⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠀⠉⠁⠀⠀⠀⠀⠀⠀⠀⠀⠄⠂
⠀⠀⠀⠀⠶⠋⠲⠶⠀⠀⠀⠠⠲⠆⠱⠧⠀⠀⠀⠀⠀⠤⠠⠴⠀⠀⠸⠄⠠⠄⠀⠀⠀⠀⠀⠶⠀⡄⠰⠄⢠⠀⠀⠀⠀⠀⠀⠸⠂⠱⠆⠆

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⢀⣀⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡿⠁⠈⠙⠛⠿⠿⠿⠿⠛⠋⠉⠙⠻⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⠁⠀⢀⣀⡀⠀⠀⠀⠀⣀⣀⡀⠀⠀⢿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠼⡇⠀⠀⠉⠉⠉⠀⠀⠀⠀⠉⠉⠉⠀⠀⢸⡿⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠘⡇⠀⠀⠀⢀⣤⠀⠀⠀⠀⣠⡄⠀⠀⠀⠀⡓⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢣⠘⡇⠀⠀⠀⠘⠟⠀⠀⠀⠀⠙⠃⠀⠀⠀⠀⡇⡠⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢃⠀⠀⠀⠀⠀⠀⠈⠉⠀⠀⠀⠀⠀⠀⢠⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣄⠀⠀⠀⢶⡠⠤⠤⠄⡲⠂⠀⠀⢀⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠈⠒⠐⠉⠀⠀⠀⡠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢢⣀⠀⠀⠀⠀⣀⡴⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠀⠉⠐⠒⠉⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠴⢇⡀⠀⠀⠀⠀⠀⡨⠦⠄⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠤⠐⠊⠉⠀⠀⠀⠀⠈⠓⠠⠤⠒⠉⠀⠀⠀⠀⠉⠑⠒⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠁⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⠈⠲⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠇⠀⠀⢘⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⡜⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⢣
⠀⠀⠀⠀⠀⠀⠀⢀⠃⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠘⡀
⠀⠀⠀⠀⠀⠀⠀⡘⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⢃
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠉⠁⠉⠉⠈⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡤⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣴⠓⠋⠉⠀⠈⠙⠒⢄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡽⠃⢀⠀⠀⠀⠀⠀⠀⠀⠀⠉⢆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡞⠀⠀⡜⢦⣀⣐⣠⣤⡼⣃⡤⠀⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠱⡀⡸⠁⠀⠀⠀⠀⠀⠀⠉⢳⠀⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡛⠃⠀⢿⠇⠀⠀⠰⡿⠀⠈⠾
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠔⠘
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⢟⠵⠛⣶⢤⢄⢀⡤⡔⠾⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⡷⠋⢠⢀⠋⠑⢶⡴⠒⢣⢸⠈⢧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢷⠗⢄⡼⣘⠀⠀⠈⠀⠆⠻⣼⡦⢶
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠏⢀⢾⣧⠇⠀⠀⠀⠀⠂⠚⢿⣇⠈⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡸⠀⣾⢄⡿⠀⠀⠀⠀⠀⠀⠀⢸⣽⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣁⣀⡨⢠⣧⣀⠀⠀⠀⠀⠀⣀⣤⡞⡜
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠛⢿⠈⠉⠉⣉⡋⠉⠉⠊⠀⠩⠠⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠈⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡀⠀⢰⠿⣇⠀⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣯⡿⠀⡿⣚⣣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡆⠉⢑⣤⣷⠋⠉⢢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠉⠉⠉⠉⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠂⠉⠉⠙⠓⠦⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣿⡇⠀⢀⣀⣀⡀⠀⠊⢣⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢹⢜⠇⠀⠀⠀⠀⠉⠳⣬⣵⣿⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡈⠈⠀⠀⠀⣀⠀⠀⠀⠈⢻⣾⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⢿⠃⠀⠀⠀⣤⣨⠷⠀⠀⠀⢠⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⡤⣶⠞⡟⠀⠈⡀⠀⠀⠀⠀⠁⣤⣠⠽⠾⠿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⠟⠋⠀⠀⠀⡤⠀⠃⠀⠀⠀⠀⢸⡢⠐⡂⠜⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠟⠁⠀⠀⠀⢀⠊⢀⣤⣘⡄⢀⣼⡓⠚⠉⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠟⠀⠀⠀⠀⣠⠀⠀⠀⠀⠀⢁⠘⠋⠉⠀⠈⠀⠀⠀⠀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡌⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⣀⣤⣀
⠀⠀⠀⠀⠀⠀⠀⢠⠧⠂⠀⠀⠀⢀⣿⠀⠀⠀⠀⠀⠀⠀⠀⢻⡀⠜⠁⠀⠈⠠⣠⣠⡤⠤⠄⠀⠀⢠⠀⢀⡀
⠀⠀⠀⠀⠀⠀⢠⠏⠀⣀⣀⣤⣤⣿⣿⡆⠀⠀⠀⠀⠀⢀⠄⠉⠃⠐⠀⠀⠀⠱⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉
⠀⠀⠀⠀⠀⠀⣾⠀⠀⠉⠁⠀⠀⠀⠀⠉⠁⠃⠀⠒⠈⠀⠀⠠⠈⠛⢷⣄⣀⣠⣀⠀⠀⠀⠀⠀⠀⠀⠀⡦
⠀⠀⠀⠀⠀⠀⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⣹⣑⠈⠉⠉⠙⠛⠒⠒⠶⠒⠒⠋⠁
⠀⠀⠀⠀⣀⠀⠈⠛⠛⠒⠂⠠⠤⠀⠀⠠⠤⠄⣈⡿⠛⠉⠉⠉⠀⠀⠀⠈⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠛⠷⢤⡢⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣿⡦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠐⠠⢤⣀⣀⣄⣀⡀⠀⠀⠀⠀⢀⣰⠿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠻⣿⣶⡶⠖⠋⠁


⣿⣿⣿⣿⣻⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣟⣻⣛⣿⣟⣛⣻⣻⣛⣛⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒⠢⣄⣙⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠈⠀⠀⠉⠓⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⢈⡀⠀⢀⣑⠆⠀⠳⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠛⠃⠀⠙⠋⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢠⣀⠀⠀⠀⢀⠠⠀⠀⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⡀⠀⠈⠉⠉⠉⠀⠀⢀⠌
⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠢⠤⢤⠤⠤⠂⠁
⣇⠀⢸⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣴⣆⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤
⠈⠁⠁⠈⠑⠦⣄⡀⠀⠀⠀⠀⠀⢀⡠⠔⠊⠀⠘⢿⣿⠃⠉⠒⠤⣄⡀⠀⠀⠀⠀⠀⠀⢀⢀⡀⡌⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠙⠒⠄⠒⠈⠁⠀⠀⠀⠀⠀⢀⣧⠀⠀⠀⠀⠀⠉⠑⠒⠒⠂⠈⠉⠀⠀⠀⠈⠒⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⠟⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠈⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⢱
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠈⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡇⠀⠀⠀⠘⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘⠄⢀⡀
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⣭⣝⣁⣀⣀⣀⣀⣈⣒⣒⣁⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⣋⣛⣉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢇⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠤⠀⠐⠒⠒⠒⠂⠸⠿⣍⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠊⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠢⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣠⠄⠀⣠⠦⡀⠀⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢇⣀⣀⣔⠤⢊⣉⠤⠚⠁⠀⠑⢦⣘⣳⢄⣀⣀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⡍⠒⠉⠉⠂⠀⠀⠀⠐⣋⡉⠙⠀⠍⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠄⠖⣰⠀⠈⣴⣶⣷⠀⠀⠀⢸⣷⣦⡉⠀⢰⡶⠂⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢐⡇⠀⠀⠻⠿⠏⢀⠀⠀⠘⠿⠿⠃⠀⠀⣷⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠤⠼⡤⠈⣁⠀⠀⠀⠀⣤⡠⠤⠤⢠⡆⠀⠀⠀⢀⡍⠠⠴⠥⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠈⡄⠀⠀⠀⠙⢾⡓⣛⠮⠁⠀⠀⠀⡌⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠈⠢⡀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠊⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠈⠒⠠⡤⠄⠤⢤⠐⠋⠀⠀⠀⠀⠀⠀⠀⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⢀⣠⡖⠒⠒⠒⠒⠂⠈⠉⠁⠢⠔⠈⠉⠁⠐⠒⠒⠒⠒⢲⣄⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⠈⠀⠀⠈⠉⠀⠀⠀⠀⠚⠛⠚⠛⢻⠒⠂⠀⠀⠈⠁⠀⠀⠀⢻
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠠⠄⢼⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢩⡰⢀⠀⠀⠀⠀⠀⠀⠀⠠⣛⡓⠛⣻⠀⠀⠀⠀⠀⠀⠀⠀⡀⢾⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉⠈⠀⠀⠀⠀⠀⠀⠀⠀⡯⢸⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣇⠻⠀⠀⠀⠀⠀⠀⠀⠀⢰⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠟⣸⠃
⠀⠀⠀⠀⢀⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⡷⠀⠀⠀⠀⠀⠀⠀⠀⠠⣄
⠀⢀⡠⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠒⠀⠀⠠⠤⠤⠴⣈⢃⣊⡸⠤⠤⠄⠀⠀⠐⠒⠒⠂⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⣀
⡖⠓⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠂⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠓
⡇
⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋

⠀⠀⠀⠤⢤⡤⣤⠤⢤⡤⣤⢠⣤⣀⣀⣠⡤⠤⠤⠤⠤⠤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄⠤⠤⠤⠤⠤⢤⣄⣀⣀⡤⢠⡤⣤⠤⢠⡤⣤⠠⠆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡇⠀⠀⠀⠀⠀⠊⠉⠉⠉⠉⠉⠉⠉⠉⣿⠀⠀⠀⠀⠀⢰⠁⠀⠀⠀⢰⠀⠀⠀⢰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢱⡆⠠⠤⢤⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢢⡤⠤⠄⢲⠊⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠇⠀⠀⢰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⣸⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⡿⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠃⠀⠀⠀⠀⠀⢸⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣈⠷⣴⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⣿⣦⠾⣇⠀⠀⠀⠀⠀⢸⠀⠂⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠞⠁⣶⡌⣌⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⠀⢇⢱⣶⡈⠳⣄⠀⠀⠀⢸⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠰⠉⠓⠀⠀⢀⠞⠉⠐⢤⠟⡰⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢆⠹⡄⠂⠉⠳⠀⠀⠘⠉⢱⠀⠸⠀⠂
⠀⠀⠀⠀⢀⠦⢦⠀⠰⠀⠄⠀⠀⠸⠃⠀⠀⠀⡴⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣦⠀⠀⠀⢀⠇⠀⠠⠄⠼⠀⠰⠦⢆
⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠒⠒⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠒⠒⠁⠀⠀⠀⠀⠀⠀⡆⠀⢘
⣤⣤⣤⣤⣴⣶⣶⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣴⣶⣾⣤⣤⣤⣤
⣿⣿⣿⠉⠈⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣿⣿⣿⣿⣷⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣾⠿⣿⣿⣶⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡾⠋⣁⠂⠉⠹⢿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠊⠀⠀⢀⠀⢀⠀⠜⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⢀⠀⠀⠈⠀⠘⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠄⠆⠀⠒⡢⠋⠢⣀⣀⡀⠤⠄⣺⣷⣿⠟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡐⠀⠀⣄⠀⠀⠑⠤⣀⢸⠁⠀⡤⢾⠟⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠁⠀⠀⠸⢧⡀⠀⠀⠀⢸⠀⠀⢣⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠀⠒⠂⠈⢳⡄⠀⠀⢸⠀⠀⠈⢆⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠢⠄⠔⢛⡢⣄⢨⠀⠀⠀⠸
⠀⠀⢀⠀⠠⣤⣤⠤⠤⠤⠤⢀⣇⡀⠀⠀⢠⠠⢐⣸⣯⠬⠆⣀⣀⡜⠀⠀⠀⠀⠀⠀⣀⣀⡀
⠀⠀⡌⠴⢉⣀⣀⡀⠠⠤⠤⠀⢈⣓⣌⢀⣀⡈⠉⠁⠈⠠⠤⠤⠍⢢⡄⠀⠀⠀⠀⣀⡀⠀⠈⠑⢄⡀
⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠤⣔⠦⠝⠂⠁⠀⠀⠀⠀⠈⠢⢀
⠀⣌⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣘⣋⣉⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣑⣤
⠀⠓⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠀⠒⠒⠐⠒⠒⠒⠒

⠀⠀⠀⠀⠀⠀⣴⣆⣤⣄⠀⠀⠀⠀⠀⠀⢠⣶⡄⠀⠀⢀⣤⣄⠀⠀⠀⠀⠀⠀⣴⣦⣤⣄
⠀⠀⠀⠀⠀⢠⣿⣅⣽⣿⡆⠀⠀⠀⠀⠀⣤⣿⡅⠀⠀⢈⣿⣽⡇⠀⠀⠀⠀⣠⣽⣧⣽⣯⣷
⠀⠀⠀⠀⠀⣸⣿⣯⣿⣿⡷⠀⠀⠀⢀⡾⢻⣿⡿⣆⣰⢟⣿⡏⢧⡀⠀⠀⠰⣿⣿⣿⣿⣿⣦⡠
⠀⠀⠀⠀⠀⣼⣿⢀⣿⣿⣷⠀⠀⠀⠉⠀⣸⣿⡇⠈⠁⣸⣿⣿⠀⠁⠀⠀⠀⢸⣿⣿⢸⣿⣿⡅
⠀⠀⠀⢠⣴⢿⣿⠀⣿⠹⣇⠀⠀⠀⠀⢀⣿⣿⡇⠀⠀⠈⢿⡏⠀⠀⠀⠀⠀⢸⣿⣿⠀⣿⣧⡀
⠀⠀⠀⠈⠁⠘⠏⠀⠻⠀⠻⠂⠀⠀⠀⠸⠃⠻⠃⠀⠀⠀⠻⠂⠀⠀⠀⠀⠀⠘⠟⠿⠀⠟⠈⠉

⠀⠀⠀⠀⠀⢰⣶⣤⣶⣄⡀⠀⠀⠀⠀⠀⣀⣴⣦⠀⠀⣴⣶⠀⠀⠀⠀⠀⠀⣠⣴⣦⣴⣶
⠀⠀⠀⠀⢠⣾⡏⢈⣽⡼⠇⠀⠀⠀⠀⠀⣹⣾⣯⠀⠀⣼⣿⣦⠀⠀⠀⠀⠀⡿⣬⣍⠈⣿⣦
⠀⠀⠀⠀⣾⣿⢷⣠⣿⣧⡀⠀⠀⠀⠀⠀⡏⣽⣿⢧⡼⢿⣿⣿⡇⠀⠀⠀⠀⣠⣿⣧⣰⢿⣿⡆
⠀⠀⠀⠀⣿⡇⠀⠁⣿⣿⡇⠀⠀⠀⠀⠀⢰⣿⣿⡆⠀⢸⣿⡏⠃⠀⠀⠀⠀⣿⣿⣏⠁⠀⣿⡇
⠀⠀⠀⠀⣿⡇⠀⠀⢸⣿⡀⠀⠀⠀⠀⠀⠀⣿⣿⠀⠀⢸⣿⠇⠀⠀⠀⠐⠶⠾⣿⠀⠀⠀⣿⡇
⠀⠀⠀⠀⠛⠃⠀⠀⠘⠛⠃⠀⠀⠀⠀⠀⠀⠛⠛⠀⠀⠘⠛⠀⠀⠀⠀⠀⠀⠀⠛⠀⠀⠀⠛⠃

⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⢀⣀⡀
⠀⢀⠀⢀⠀⢀⣤⠀⠀⠀⢠⣾⣿⠿⠿⣇⠀⠀⠀⡔⠁⠀⠉⠂⠀⠀⠀⠠⣊⠉⢄⠀⠀⠀⠀⢠⣾⠿⠿⣿⣆
⠀⡈⠴⡖⢠⡞⢭⠀⠀⠀⡿⠏⠀⠀⠀⢿⠀⠀⠀⠇⠐⠀⠂⠠⠀⠀⠀⣾⣿⡿⣿⡷⡆⠀⢠⣷⠋⠀⣀⠘⡿⠄
⠀⠈⠈⢁⣠⠉⢘⠁⠀⠀⢧⠀⢀⡀⠀⡻⠀⠀⠀⢀⢐⠋⢂⠀⠀⠀⠀⢳⣾⡛⢣⡾⠀⠀⠘⣍⠉⠀⠀⠀⣠
⠀⠀⢣⣀⠀⣠⠆⣠⠀⠠⣾⡆⠀⠄⣰⠋⠀⠀⣀⣼⡁⠠⠈⣆⡀⠀⠀⠸⡿⣿⡽⠱⠂⠀⠀⠜⣧⠀⠀⠀⠏⠁
⠀⠀⠀⠙⠛⠁⠈⠉⠀⠀⠙⠳⠄⠚⠋⠉⠀⠀⠉⠁⠙⠶⢺⡏⠁⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⢀⡠⠤⠤⠀⠀⠀⠀⠀⣀⣀⠀⠀⠀⠀⢀⣤⣤⡀⢀⣠⣤⣀⠀⠀⢀⠀⠀⢠⡀
⠀⠀⢠⣾⠟⠉⠉⢳⡀⠀⠀⢀⣿⣤⣄⣤⣦⡀⠀⠀⣾⠁⠀⠉⣷⠀⠠⣿⣽⣿⡿⠛⠛⢿⣿⠆⠀⣧⠁⠀⠀⠸⠄
⠀⠀⠀⠸⡺⠁⠰⠛⢧⠀⠀⢾⡟⣻⣏⣿⡴⠇⠀⢸⠇⠀⠀⠀⢻⠀⠀⠻⣻⡟⠀⣀⠀⡈⠁⠀⠀⠁⠀⠀⠀⠐
⠀⢠⠀⢸⠈⠁⠀⢀⠏⠀⠀⠀⣯⡷⣤⣴⡃⠀⠀⢸⣷⣄⠀⢠⣾⠀⠀⠀⠀⢠⠀⡡⠤⠄⠀⠀⢀⣅⠀⠀⠀⢢
⠀⠸⠀⠈⠓⠀⢰⠊⠀⠀⠀⠘⢯⣿⣶⢞⣅⠀⠀⠈⣽⠟⠀⠀⠁⠀⠀⠀⠀⠲⠗⠤⠐⠀⠀⠀⠉⠀⠀⠠⢶⠣
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠈⠉⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠔⠢
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣤⡜⠦⠼⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢙⢱⣄⣀⣀⣀⣀⡀⠠⠤⠤⠤⠀⠀⠀⠀⠀⠀⣒⣒⠒⡒⢒⣒⠒⣒⣂⡀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⡠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⣕⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢤⡖⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢞⠁⣆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⠁⠈⠉⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⢏⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠣⣀⠀⠀⢀⠏⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⢉⢂⠀⠀⠀⠀⠀⠉⠉⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠡⠈⡎⡀⠀⠀⠀⠀⠀⠀⢀⠤⠄⠀⠾⢶⣦⡀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠃⡠⠭⡀⢣⠀⠀⠀⠀⢠⠞⠁⡀⠀⠀⠀⠀⠀⠙⢦
⠀⠀⠀⢀⠀⠀⠀⣠⠐⢀⠁⠀⢤⣽⣒⠶⣶⡄⠀⡦⠀⠀⢘⢸⠁⠀⢃⢸⠀⠀⠀⠀⠃⠀⠀⠠⠀⠀⠄⠀⠀⠀⠈⠇
⠀⠀⢠⡁⠀⢾⡟⠐⠊⠁⠀⠀⠀⢀⠀⠀⠘⢄⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⣄⠀⠀⠀⡄⢂⠀⠀⠀⡀⠀⠀⠀⢀⡀⡐⠠
⠀⠀⠀⠀⠀⢸⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠸⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠎⢇⠀⠀⠑⢌⠑⠒⠉⠀⠀⠀⠀⠈⡡⠃
⠀⠀⠀⠀⠀⠛⢆⠀⠀⠀⡀⠀⠀⠰⢀⠀⢀⠄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡌⣀⠀⠀⠀⠁⣢⠄⣀⣀⡤⠔⠊⠀⠀⢐
⠀⠀⠀⠀⠀⠀⠈⠳⢄⡀⠈⠐⠊⠀⡋⢄⠁⠠⠤⠄⢀⠀⠀⠀⠀⠀⠀⠀⠀⢡⠙⣦⣤⠴⢊⠛⠜⠁⢠⡃⠑⢆⠀⡀⠈⠄
⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⢨⠘⠧⠛⠀⠈⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠊⠀⠀⢸⡁⠀⠀⣀⣁⡠⠞⠐
⠀⠀⠀⠀⠀⠀⢀⠞⠁⡧⡸⠀⠀⠀⠉⠊⠀⠀⠀⠀⠀⠀⠀⠀⠀⠤⡀⠀⠀⠀⠀⠀⠀⠀⠸⠁⡉⠁⢸⠁⠀⠀⠀⠀⢀⠤⡀⠀⠀⠀⡠⡀
⠀⠀⠀⠀⠀⠀⠘⠢⠀⡷⠃⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⡤⢣⡰⠀⡀⡄⠀⠀⠀⠘⠤⠛⢤⡼⠀⠀⠀⠀⠀⠠⢀⡏⢀⡀⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⡔⢬⡀⡀⣀⡰⠀⠀⠀⠀⠀⠀⠀⠀⠪⡨⢛⢾⠇⠸⠃⡇⠀⠰⠒⠼⠀⣠⢤⡧⠀⢀⠀⠀⠀⠀⠀⢑⢀⡀⠎⠠⢄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢨⠁⠀⣁⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠏⠹⠤⢅⠼⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⢀⣇⡀⢘⡆
⠀⠀⠀⠀⠀⠀⠀⠉⠉⠁⠈⠵⠄⠈⠚⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣲⠀⠆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠤⠤⠜

⠀⠀⠀⠀⢀
⠀⠀⠀⢰⠁⢀⠀⠀⠀⠀⠀⠀⡀⢀⢀⢀⡀⢀
⠀⠀⠀⠫⠀⠓⠈⠹⠀⠁⠇⠈⠃⠈⠈⠸⠁⠋⠀⠡⠅⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⢀⣀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⡏⠛⢦⡀⣾⠁⠀⡴⠋⠉⢻⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣴⣾⣉⣉⣻⣷⣦⣷⣿⣶⠟⢓⣺⡶⢿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠿⠥⡤⠶⠿⢿⠿⠿⢿⣿⣿⣷⣾⣻⢶⣄⠘⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⠴⠋⢁⡄⠀⠀⡠⠞⢁⡤⠞⠁⠔⠊⠉⠛⢿⣦⣙⠿⠖⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⠶⠋⢓⠩⠒⡥⠊⠀⣊⠴⠚⠁⠀⠐⢀⠔⠋⠀⠀⢉⣿⡿⢯⡑⢦
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣶⣿⣽⣋⠁⠀⠁⠀⢀⢔⣶⣍⣀⡀⠂⠀⠀⠒⠁⡀⠀⢀⡴⠋⢺⣿⠈⢿⢸
⠀⠀⠀⠀⠀⠀⠀⢰⡿⢩⣴⡶⠶⠮⠳⣄⣠⠞⠁⣾⠁⠀⡈⠙⢳⣄⠀⠀⠀⠄⣰⠋⠀⠀⠀⣿⡄⣸⠃
⠀⠀⠀⠀⠀⠀⠀⠸⣷⡀⢿⣤⠂⠀⠀⢸⡇⢀⠠⣈⠛⠛⢠⠖⠀⢻⡆⠐⡠⡾⠋⠀⠀⠀⠀⢻⡷⠃
⠀⠀⠀⠀⠀⠀⠀⠀⢈⡿⢮⡻⢶⣤⣴⣞⡵⣣⠞⠑⠁⠀⡼⢧⣀⣼⣧⠊⠀⠀⠀⠀⠀⠀⣠⢾⡇
⠀⠀⠀⠀⠀⠀⢠⡾⠛⠛⢷⣭⢄⣉⠟⠩⠞⠛⠛⠻⣦⣂⡔⢁⣞⣵⠋⠀⠀⠀⠀⡠⢂⣼⣵⠟
⠀⠀⠀⠀⠀⠀⢸⡇⢀⡶⣧⡹⠋⣁⠀⣴⠟⠛⢻⢦⡈⢿⡄⣸⡁⠈⢳⡄⠂⠀⠊⡠⠋
⠀⠀⠀⠀⠀⠀⠸⣇⠸⡇⠘⠃⢺⡅⣠⣿⣠⠀⠀⣸⠇⣼⢏⣴⡓⠒⠀⣧⠀⢠⠞
⠀⠀⠀⠀⠀⠀⠀⠙⠦⠙⠦⣀⣼⡟⠁⠙⠿⣶⣬⣤⡾⠏⢸⠀⢻⣀⢴⣏⠔⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠣⣟⡥⠞⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⢀
⢀⠀⠀⠀⠰⢄⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⣁⣠⣖
⠀⠀⠀⠀⠀⠀⠈⠑⠆⠒⠒⠉⠁⠀⣀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠁
⠀⠀⠀⣤⣀⣀⣡⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣾⠿⠿⣿⡟⠛⠛⢻
⡄⠀⠀⣿⠟⠛⠛⠛⠉⠁⠁⠉⠁⠀⠀⠀⠀⠀⠙⠃⠀⣿⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⣸
⡇⠀⠀⣿⡀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡿⢦⡀⠀⠀⠀⠀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣌
⠇⠀⠀⣿⠙⠛⠛⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠀⠀⠀⠉⠉⠉⠉⠉⠉⠉⠉⠙⠉⠉⠉⠉⠈⠁⣿
⠄⠀⠀⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⡆⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⣿
⡇⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⠀⠀⡇
⠃⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄⠀⠀⠀⡇
⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇
⠀⠐⣸⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠃

⠀⠀⠰⣿⣷⡀
⠀⠀⠀⠹⣿⣷⡀
⠀⠀⠀⠀⢻⣿⣿⣀⣀⣠⣤⣤⣤⣤⣤⣄⡀
⠀⠀⠀⢀⣤⣿⣿⣿⡏⠉⠉⠀⠀⠀⠈⠙⢿⣄
⠀⢀⣴⠟⠁⠀⢻⣿⣿⠀⠀⠀⠀⠀⠀⢀⡿⠹⣦
⢰⡟⠁⠀⠀⠀⠀⠛⠁⠀⠀⠀⠀⠀⣠⠞⠁⠀⠹⣧
⣿⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⠞⠁⠀⠀⠀⠀⠹⣧
⢹⣇⠉⠛⠶⢦⣤⣤⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⠀⢹⣇
⠀⠹⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⡀
⠀⠀⠈⠻⣦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠇
⠀⠀⠀⠀⠈⢻⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠟
⠀⠀⠀⠀⠀⠀⠹⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⡾⠋
⠀⠀⠀⠀⠀⠀⠀⠹⣷⣄⣀⣀⣀⣀⣤⣴⣾⣿⡉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠈⢻⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣷⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣦⡀⠀⠀⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠻⣿⣿⣿⣿⣦⡀⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣷⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢿⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢿⣿⣿⣿⣆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿⣿
⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣶
⣿⣿⣿⡟⠉⠉⠉⠉⠉⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⣿⣿⣿⣷⣶⣶⣶⣿⣷⣤⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠳⠤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠢⡀
⠀⠀⠀⠀⠀⡰⠀⠀⠀⠀⠀⠀⣀⣠⠄⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠠⡀
⠀⠀⠀⠘⠌⠁⠀⠀⠀⠀⠈⠃⠐⠂⠁⠀⠀⠤⠄⠐⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠶⠦⡀⠀⠀⠀⠀⣆⡙⢄⠐⠀⡀
⠀⠀⠀⠀⠅⠀⠀⠀⠀⠈⠡⠚⣁⠀⠀⠀⠀⠄⣢⡄⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡞⢡⣶⠢⢹⡆⡴⢋⡩⠉⠻⣆⠴⢋⣽⡄⠠
⠀⠀⠀⠀⡅⠀⠈⡀⠀⠠⠀⠀⢠⣿⡄⢀⡀⠀⢻⡇⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠉⢧⡸⢟⣀⡼⢻⡇⣾⡿⠀⠀⣿⣤⣉⠈⠀⠀⠂
⠀⠀⠀⠀⠠⠀⠑⠚⣢⠅⠀⠀⠨⠟⠁⠀⠉⡷⠀⠀⢀⠀⠀⠀⢐⡀⠀⠀⠀⠐⠀⠀⠀⠙⠙⠃⠀⠀⠳⢬⣀⣀⠴⠟⠀⢹⠆⣀⠀⠀⠆
⠀⠀⠀⠀⠀⠀⢅⠈⠥⡀⠀⠀⠀⠀⠦⢠⡦⠀⠀⠀⠌⠀⠀⠀⠘⣭⡀⠀⠀⠐⠠⠒⠃⠤⠀⠀⠘⠂⠛⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢘
⠀⠀⠀⠀⠀⠀⠈⠐⠀⠘⠂⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⢠⣜⡤⢢⠴⠀⠀⠀⠀⠈⠂⠀⠀⡀⠀⠀⠀⠤⡀⣀⣀⡀⡀⠀⠀⠂
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠰⢯⣤⣬⠦⠄⠠⠠⠀⠰⠀⠀⠀⠘⠭⠅⠀⠀⠀⢠⣀⡀⠀⢀⣤⢠⣅⠀⠡⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠀⢀⠀⠀⠀⠀⠀⠀⡆⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⢰⢀⠋⡍⢸⠈⣿⡄⠀⠠⠆⣼⢃⠕⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠃⠀⠀⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠃⠘⣷⡋⠀⠤⠀⠀⠛⢧⢊⠤⠆
⠀⠀⠀⠀⠀⠀⠀⠠⡀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣉⠀⠴⠀⠀⠋⢀⢨⡇⡀⣹
⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠢⠤⠤⠤⠤⠤⠤⠀⠂⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⡭⠥⠤⠤⠤⡯⢤⠌⠰⠇⠐⠆
⠀⠀⠀⠀⠀⠀⢀⠈⢀⢄⡰⢀⣀⣻⣤⣤⡀⠀⠀⠶⢒⣲⣤⠴⢶⠒⠀⠀⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⡖⠀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⣠⢥⡐⠅⠀⠁⠀⠉⠛⠛⠛⠁⠀⠀⠀⣀⣀⣀⠐⠨⠋⠉⠉⠙⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠩⠴⡮⢄⢀⡀⣂
⠀⠀⠀⢀⡄⢉⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠘⠀⠈⠉⠙⠛⠃⠀⠀⠀⠀⠀⠀⠓⠐⠒⠀⠀⠀⠒⠀⠂⠀⡁⠈⠑⠈⡋⣂
⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⢸
⠀⠀⠀⢸
⠀⠀⠀⢸
⠀⠀⠀⢸⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⢀
⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠀⠀⠀⠈
⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠

⠀⠠
⠀⢀

⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀

⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣶⣶⣿⣶⣤⣄⡀⠀⠀⠀⠀⠀⢠⠤⠤⠤⠤⠤⠤⠤⠤⠤⣤⠖⠒⠒⠒⠒⠛⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡟⠙⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠒⠒⠲⠦⠤⢤⣤⣿⡀⠀⠀⠀⠀⠀⢹
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣤⣤⣀⣀⠀⠀⠀⠀⠀⠀⠉⠳⢦⣀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⠛⠿⢿⣿⣿⠟⠓⠀⠀⠀⠀⠀⠀⠀⠉⠛⢦⣼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠶⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉





⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣯⣽⣟⣛⣛⣛⣿⣛⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠀⠀⠈⠙⠛⠂⠂⠄⡀
⠀⠀⢀⠀⠂⠁⠀⠀⠀⠀⠁⠆⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⢄
⠀⠔⠕⠀⠠⠊⠀⠀⠀⠀⠀⣴⡀⠐⡀⠀⠀⠀⠀⠀⠀⠀⠀⣆⡀⢀⣀⣀⣠⡤⠀⢀⠀⠀⠀⣀
⠈⠈⠀⠀⠀⠀⠀⠀⢀⠀⡴⠀⢓⠀⢡⠀⠀⠀⠀⠀⠀⠀⠀⠀⠐⠀⠠⠀⠘⠂⠁⠃⠄⠀⠈⠀⠡⡀
⡆⠀⠀⠀⠀⠀⡠⢀⠡⠚⠀⠀⠀⠁⡀⢦⠆⠀⠀⠀⠀⠀⠀⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠂⠁
⢃⠀⣀⠀⠠⠁⠂⠀⠀⠀⠀⢠⠀⠀⠀⢹⢉⠀⣀⢀⠀⠀⢴⡃⠀⠀⡤⠀⠀⠀⠀⠀⢀⠀⠀⠀⢰⠂
⠀⢀⡀⠀⠀⢠⠀⠀⠀⠀⠀⠀⠀⠀⠀⡘⠙⣀⠓⢄⠇⠀⠐⢂⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⡘⠑
⠀⢽⢣⠀⠀⠈⠀⠀⠀⠀⠐⠀⠀⢀⢴⣡⠖⠁⢀⠂⠀⠀⠀⠀⠁⢀⠀⠀⠀⠉⠁⠀⠀⠀⠀⠔
⠀⠀⠀⡗⠤⡀⠀⠀⠀⠀⠀⠄⡒⡟⢃⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡠⡖⠂⣀⠲⣆
⠀⠀⠀⠃⠀⠁⠀⠀⢠⢧⣀⠘⠂⠇⠀⠓⠁⠀⠀⠀⠀⠀⠀⢀⠔⠃⠀⠈⠂⠛⠊⠀⠁⠢⡀
⠀⠀⠀⠁⠀⡀⠀⠐⠀⠀⠀⠈⠀⠀⠀⠀⠂⠀⠀⠀⠀⠀⠐⢁⡀⡆⠀⠀⠀⠀⠀⠀⠧⡠⠀⠂
⠀⠀⠀⠀⠀⢀⡀⠀⠠⢆⣀⣀⠀⠄⠀⠀⠘⠠⢤⠀⠀⠐⠢⠀⢈⡁⡗⠐⠲⡀⠠⢰⣇⣣⡤⠒⠠⢤⣀⡀
⠀⠀⠀⠀⠀⠉⠻⢦⠤⠠⠄⠀⡁⠀⠀⠤⠤⢖⣀⣓⣀⠀⠀⠀⢀⣁⡒⣒⡒⣒⣚⣋⣭⣭⣥⣤⣤⢤⣶⠿⠿
⠀⠀⠀⠀⠀⠀⠀⠀⠉⠲⠖⠶⠶⠶⠒⠒⠀⠉⠙⢌⠉⠉⠉⠀⠨⡁⠉⠉⠹⣔⠈⠉⠉⠀⢤⡀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⢖⡀⠀⠈⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠐⣤⠀⠀⢠⣀⣐⡄⠀⠀⡤⠄⠸⠸⠆
⠀⠀⠀⠀⠀⠀⠀⠉⠑⠠⠆⠶⡄⠀⢀⠠⠄⠀⡖⠀⡙⠀⠀⠉⠉⠉⢁⠀⠀⠀⠀⠀⠀⠀⢠⢀
⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠐⢄⠄⡆⠀⠀⡁⠨⠁⠀⠀⠀⠀⠀⠀⠂⣀⠠⠀⠈⠒⠐⠇⠐⠀⡐⢀
⠀⠀⠀⠀⠀⠀⠀⠀⠪⡂⠄⠀⠀⠈⠢⢴⠄⠀⠘⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣤⣴⣶⣶⣶⣿⣶⣶⣶⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⠛⠻⠿⠿⠿⠿⠿⠿⣿⠿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⢫
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⠀⠀⠀⠀⡀⢀⢩⠕⡄⢸⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠟⡛⠀⠀⠀⢸⣐⣂⠗⠚⣕⢼⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣄⣉⠀⠀⠀⠀⠀⠀⠢⡠⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢦⡀⠀⠀⠀⠾⠓⠒⠿⣠⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡝⠀⠠⣤⣤⠤⠖⠋⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣠⣤⣤⣶⣶⣶⠊⠀⠀⠀⠀⠀⠀⠱⣤⣄⡀
⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣷⣄⡀⠀⠀⠀⠀⣰⣿⣿⣿⣿⣦⣄
⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⢀⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣄
⢀⣴⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿⠟⣛⠟⠛⣻⣿⣿⣿⣿⣿⣦
⠁⠈⠻⣿⣿⣿⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⢭⠢⠍⠥⢸⣿⣿⣿⣿⣿⣿⣇
⠀⠀⠀⣹⣿⡿⠟⠛⠉⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡧⠕⠪⠹⢍⣾⡿⢿⣿⣿⣿⡿⠃⢳⡀
⠀⠀⠺⡀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣮⣿⣿⣿⡇⠀⠙⢛⠧⠄⠀⢘⡇
⠀⠀⠀⠘⢦⡀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⡞⠀⠀⠀⣸⠃
⠢⡀⠀⠀⠀⠙⣄⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⣼⠁⠀⠀⣰⠏
⠀⠈⠢⣄⠀⠀⠈⠙⢦⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠇⢰⠃⠀⠀⣼⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠉⠀⠀⠈⠀⠀⠘⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣷⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⣿⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣠⣤⣤⣤⣤⣤⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠛⠛⠛⢻⣿⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⣿⣿⡿⠿⠿⠿⣿⣿⣷⣦⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⠟⠉⠀⠀⠀⠀⠀⠀⠉⠻⣿⣿⣄⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⣿⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⠀⠀⠀⠀⠀⠀⠀⢸⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡆⠀⠀⠀⠀⠀⠀⣿⣿⡇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠀⠀⣸⣿⣿⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣧⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⠃⠀⠀⠀⠀⣰⣿⣿⡏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⣷⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⠏⠀⠀⠀⢀⣼⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣿⣿⣿⣶⣤⣤⣤⣤⣤⣴⣾⣿⣿⣿⠋⠀⠀⠀⣠⣾⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⢿⣿⣿⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⣦⣤⣾⣿⣿⣿⠏
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⣉⣉⣥⣤⣤⣿⣿⣿⣿⣿⣿⣿⠛⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣶⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣾⣿⡿⠿⠛⠋⠉⠉⠀⠀⠀⠀⠀⠈⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣶⣿⡿⠛⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣾⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢻⣿⣿⣿⡆
⠀⠀⠀⠀⠀⠀⠀⠀⣴⣿⠟⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣿
⠀⠀⠀⠀⢀⣠⣤⣾⣿⡁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⣿⣿⣿⡇
⠀⠀⠀⣰⣿⣿⣿⣿⣿⣿⣆⠀⠀⠀⠀⠀⠀⠀⠀⢀⣴⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣧
⠀⠀⠀⢹⣿⣿⣿⣿⣿⣿⣿⠆⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢹⣿⣿⣿
⠀⠀⠀⠀⠙⠿⢿⡿⠋⠀⠀⠀⠀⠀⠀⣠⣾⣿⣿⣿⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣴⣿⣿⣿⣿⣿⡿⢿⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⠀⢀⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣤⣾⣿⣿⣿⣿⣿⡿⠛⠁⠘⣿⣿⣿⡆⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⡟
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⠿⠛⠉⠀⠀⠀⠀⢹⣿⣿⣷⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣀⠀⠀⠀⠀⠀⠈⣿⣿⣿⡆⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⣷⣦⣄⡀⠀⢹⣿⣿⣷⠀⠀⠀⠀⢀⣿⣿⣿⣿⣿⣿⣷⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣷⣦⣿⣿⣿⡆⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠿⢿⣿⣿⣿⣿⣿⣿⣄⠀⣼⣿⣿⣿⠏⠹⣿⣿⣿⣿⣇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⠀⠀⠈⠙⠻⢿⣿⣿⣿⣿⣿⣿⣿⣏⠀⠀⠹⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠹⣿⣿⣿⣿⣿⣿⣿⣿⣷⣶⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⠀⠀⠈⢿⣿⡿⠋⠉⠛⠻⠿⠿⣿⣿⡿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⠃
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠿⠿⠟⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠚⢉⡛⢦⡀⠀⠀⣀⣤⣤⣄⡀
⠀⠀⠀⠀⠀⠀⢀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡀⠀⠀⣿⠀⢠⡏⠙⠀⡧⠀⣼⠋⢠⡞⠧⠙⡆
⠀⠀⠀⠀⠀⠀⣏⢼⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⢋⡽⣯⠻⡄⠹⣦⡀⠙⠒⢛⣁⣀⣿⡀⠈⠳⠦⠾⢣⡴⠖⠶⢦⡀
⠀⠀⢸⢉⢧⣀⣿⡀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠁⠘⢧⣬⡴⠃⢀⣨⠿⠛⠋⠉⠉⠉⠉⠙⠓⠶⣄⡀⡏⠀⣴⠶⣆⣿
⠀⠀⠈⠋⡼⠩⠀⠈⡇⠀⠀⣠⠖⢋⣉⠙⢦⠀⠀⠀⠀⠀⠀⠘⢧⣄⡀⢀⣀⡴⠋⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠙⢧⠀⠹⢦⣥⠟⠀⠀⠀⠀⠂
⠀⠀⠀⠀⠙⠦⢤⡞⠁⢀⠞⠁⠀⢾⡿⠃⡜⠀⠀⠀⠀⠀⣀⣀⡀⠉⠉⢉⡟⠁⠀⠀⠀⢀⣄⠀⠀⠀⠀⢀⡀⠀⠀⠀⠈⢷⡀⠀⢀⣀⡀⠀⠀⠀⠃
⠀⢀⣀⠤⠤⠤⣄⣷⣠⠏⠀⠀⠀⠀⠀⠚⠓⢢⠀⠀⢠⡞⣩⠎⢹⡆⠀⣼⠀⠀⠀⠀⠀⠈⠛⠀⠀⠀⠀⠘⠃⠀⠀⠀⠀⠈⢷⣴⠋⣉⣉⢳⡄
⡔⣩⣤⡀⠀⠀⠀⠈⢧⠀⠀⠀⠀⠀⠐⠿⠟⣠⠇⠀⢸⡄⠘⠓⠛⠀⣰⣟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⢸⠃⢼⠃⣷
⢇⡻⠟⣃⡄⠀⠀⠀⠈⣿⠑⠒⠒⠒⠒⠒⠊⠁⠀⠀⠈⠳⢤⣤⡤⠶⠋⢿⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡶⠀⢸⠇⠈⠳⠤⠞⠃⠀⠄
⠀⠁⠉⡏⢰⣿⣷⢀⡞⠉⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠲⠶⢤⣤⣄⣀⣀⣠⣤⠴⠶⠛⠋⠀⣠⠏
⠀⠀⠀⠘⠶⠭⠴⠋⠀⠀⠀⢧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⢦⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⡾⠃
⠀⠀⠀⠀⠀⠀⢰⡟⠓⠶⣤⡀⢀⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣼⠛⠲⠤⣤⣤⣤⣤⡤⠴⠖⠛⠁
⠀⠀⠀⠀⠀⠀⠘⢧⠀⠀⠀⠙⠇⠀⡼⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⡿⠁⠀⠀⠀⠀⠀⠀⢰⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠙⠢⠤⠤⣤⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡴⠋⢰⠃⠀⠀⠀⠀⠀⠀⢠⠎⢣⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢺⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡤⠞⠁⠀⢠⠇⠀⠀⠀⠀⠀⢀⣴⠋⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣦⣤⣤⣤⣤⡤⠴⠖⠋⠁⠀⠀⠀⠀⡞⠀⠀⠀⢀⣠⠴⠋⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠳⣤⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣰⣁⣤⠴⠞⠋⠁⠀⠀⠀⠀⠀⠀⢸⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠙⠛⠲⠶⠶⠖⠒⠛⠉⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢷
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢳⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⢠⣶⣤⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠲
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠈⠛⠋⠀⠰⣿⣷⠀⢀⣶⣄⠀⢀⣀⠀⠀⢠⣦⣄⢻⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣼⠁⠀⠀⠀⠀⠀⠈⠁⠀⠘⠛⠁⠀⠻⠟⠁⠀⠙⠛⠁⠈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡏⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠹⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠛⠓⠒⠲⠶⠶⢶⡶⠶⠶⠶⠶⠶⠶⠶⡶⠶⠶⠶⠒⠚⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠸⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡇⠀⠀⠀⠀⠀⠀⠀⠀⢷⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠒
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠸⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣷⠀⠀⠀⠀⠀⠀⠀⠀⠀⣧
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣧⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠋
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⠖⠛⠛⠛⠲⢦⣿⠀⠀⠀⢀⡴⠚⠋⠉⠉⠉⠙⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠸⢧⣀⣀⡀⠀⠀⢀⣽⠀⠀⠀⢸⣁⠀⣀⣀⡤⠶⠚⠁

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⢠⣾⠿⠿⠿⠿⠿⣷⡄⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⣿⣿⣿⣿⣿⣿⣿⣧⡀
⠀⠀⠀⠀⠀⠀⠀⣹⠃⠀⠠⠀⠠⠀⠈⢳⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⡿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠀⠀⠀⠀⠈⠂⠐⢄⡀⣀⠔⡠⠂⠀⠀⠀⠀⢀⠀⠀⠀⠿⣿⠇⠀⠄⠀⠨⠉⢻⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⡠⠛⢂⡲⣀⠛⠢⣀⠀⠀⠀⡰⠉⠁⠀⠀⠀⠀⠢⡐⠄⣀⠠⢂⠔⠀⠀⠀⠀⢀⡀
⠀⠀⠀⠀⠀⠀⠀⡔⠀⢠⠀⠀⠀⠀⡄⢀⠧⢀⠎⠀⠀⠀⠀⠀⠀⠀⠀⢀⠝⠀⠈⠑⢄⠀⠀⠀⢀⠌
⠀⠀⠀⠀⠀⠀⠀⠩⠓⢡⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡼⠐⠀⠒⠀⠆⠑⢄⠀⠂
⠀⠀⠀⠀⠀⠀⢀⠃⠀⢸⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢰⠃⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⢀⠂⠀⠀⢸⢠⠄⠭⡭⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠑⠀⠀⠀⠀⠀⠀⠁
⠀⠀⠀⠀⠀⠛⠀⠀⠀⢸⠊⠀⠀⠈⢚⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡆⡀⠀⠀⠀⠀⠀⠀⠰
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⡄⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠱⠀⠀⡃⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠉⠙⠉⠉⠙⠉⠉⠉
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠠⠀⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⠀⠀⠀⢀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⢀⣀⠀⢈⣀⢨⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡆
⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠹⠀⠀⠀⠸⠰⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⡲⠀⠀⠀⢖

⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠛⠛⠻⢿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠁⠀⠀⠀⠀⠀⠙⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠐⠢⡄⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⣫⣭⡛⣛⣛⠛⣛⣛⣛⣛⣛⣛⣛⣛⣛⠛⣛⣛⣛⣭⣝⣿⡿⠛⠿⣿⣿⣿⣿⣿⣿⣷⡀⠀⠀⠀⠀⠀⣰⣼⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⠙⡟⣥⣭⣭⣤⣭⣭⣭⣭⣭⣭⣭⣭⣭⣤⣭⣭⣬⣟⣯⣀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣦⣤⣤⣴⣾⣿⣿⣿⣿⣿⣿⣿⣿
⠀⣿⣿⣿⣿⠀⠇⢿⣿⣿⣿⣿⣿⣿⣿⠟⠛⠛⠛⠛⠛⠛⠛⣿⣿⠋⠀⠀⠀⠀⠀⠙⢿⣿⣿⣿⣿⣿⣿⣿⣿⠿⠿⠿⠿⠿⠿⠿⣿⣿⣿
⠀⣿⣿⣿⣿⠀⡆⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡼⠃⢀⠀⡀⠀⣀⣀⣀⣈⢿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⣿⣿⣿⣿⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠛⢻⣿⣿⣿⣿⣿⣿⣿⣽⣿⣿⣿⣿⡿⠁⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⣿⣿⣿⣿⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⣿⣿⣿⣿⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠟⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⣿⣿⣿⣿⠀⡇⣿⣿⣿⣿⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⣿⣿⣿⣿⠀⡇⣿⣿⣿⣿⣿⠟⠋⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⡿⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿
⠀⣿⣿⣿⣿⠀⡆⣾⣿⡿⠛⠁⠀⠀⠀⠓⠒⠒⢲⡞⠒⠒⠚⡇⠀⢸⣿⣿⣿⣿⣿⣿⣿⣷⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⣿⢿⣿
⠀⣿⣿⣿⣿⣠⣧⣠⣤⣤⣤⣤⣤⣬⣬⣀⠀⠀⠀⠁⣀⣴⣶⣷⣦⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣠⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣭
⠀⣿⣿⣿⠟⢻⡿⠻⠿⠿⠿⠿⠿⠿⠿⠿⠀⠀⠀⢸⠁⣿⣿⣿⣿⠀⠁⠀⠀⠀⠀⠀⠈⢸⣿⣿⣿⣿⣷⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠿⣋⣁⣀⣸⣿⣶⣶⣶⣶⣶⣶⣶⣶⣶⣶⠀⠀⠈⠒⣿⣿⣿⣿⠀⠀⠀⠀⠀⠀⠀⣀⣸⣿⣿⣿⣿⣻⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣶⣶⣾⣶⣿⣿⣿⣿⣶⣶⣶⣶⣶⣶⣶⣿⣿⣿⣿⣿⣿⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⡍⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⣀⣀⣀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠉⠉⠉⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⢿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⡇⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣠⠇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⠘⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡿
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⢤⣤⣤⣤⡤⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣷⣶⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣷
⠀⢸⣿⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠘⠛⠃⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠚⠁⠀⠀⠀⠀⠀⠀⠀⠀⠛⠛⠙⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠛⠋

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡠⠤⠴⠊⠉⠙⢢⡀⠀⠀⢠⠖⠈⠑⠢⠤⢄⡀
⠀⠀⠀⠀⠀⠀⠀⡠⠒⠉⠁⠀⠀⠀⠀⠉⠓⠢⣀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠀⣀⡾⠘⠊⠉⠉⠀⠈⠉⠉⠒⠔⢢⠀⠀⠀⠱
⠀⠀⠀⠀⠀⢠⠎⠀⣀⣀⣠⠖⠉⢷⠒⢒⣐⣐⡚⠳⣄⠀⠀⠀⠀⠀⢀⣘⠔⠁⠀⠀⠀⠀⠀⠀⢀⠔⠢⡀⠈⠃⠰⡀⠀⠘
⠀⠀⠀⠀⢀⠇⠰⡏⠀⠉⡤⠦⣀⣸⠉⠀⠀⠁⠀⠙⢍⢆⠀⠀⠀⠀⡴⠀⠀⠀⠀⠀⠀⠀⠀⠀⡎⠀⣀⠈⡒⠠⢄⠱⡀
⠀⠀⠀⠀⢸⠀⠀⠙⢄⠜⢰⣄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢳⠆⠀⠀⠘⣆⡀⠀⠀⠀⠀⠀⢀⡀⢈⡡⠘⠛⠀⠀⡰⠁⡧⠔
⠀⠀⠀⠀⠸⠀⠀⠀⠀⠀⢸⠀⠑⠚⠒⢤⣀⠀⣀⠀⠀⠀⣔⠃⠀⠀⠀⠀⢸⠙⠉⠉⠉⠉⠁⠍⠀⠀⢸⠀⠑⠈⠀⠀⡇
⠀⠀⠀⠀⡆⠀⠀⠀⠀⠀⡞⠀⠀⠒⠀⠀⠀⠉⣉⠈⡙⠉⠁⠀⠀⠀⠀⠀⠀⠐⠂⠀⠀⠀⠀⠂⠀⠀⢸⠀⠀⣀⠀⡰⠁
⠀⠀⠀⢠⠃⠀⠀⠀⠀⢠⠃⠀⠀⠂⠀⠀⠀⠠⠌⠰⠀⠀⠀⠀⠀⠀⠀⠀⠈⢄⠀⠠⣀⡠⠄⠀⠀⠀⠀⠉⡉⠜⠉
⠀⠀⠘⠣⡄⠀⠀⠀⣴⢧⣀⠀⠀⠐⠢⠠⠀⢀⡠⠃⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠢⠌⣀⣀⡀⠀⢤⣔⡊
⠀⠀⠀⠀⠙⠚⢢⡼⠀⢀⠑⠭⠶⣦⣒⣂⡬⠇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡰⠉⠳⠴⠋⡀⠀⠂
⠀⠀⠀⠀⠀⠀⡘⡄⠀⠘⠀⠀⠀⡁⡄⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠠⠀⠀⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⡀
⠀⠀⠀⠀⠀⠀⡇⢰⠀⠀⡆⠀⠀⠁⠰⢄⡀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠂
⠀⠀⠀⠄⠀⠘⠄⠀⢆⠀⠉⠂⠄⣗⠄⣀⠀⠑⠢⢄⣀⠂⠀⠀⢀⠀⠀⠀⠀⠈⠁⠀⢐⠄⠀⠀⠀⠄⠈⠀⢀⠄
⠀⠀⢨⠀⠀⠀⠈⠁⠀⠑⠢⠤⢀⡀⠉⠙⣢⡰⢄⣠⠏⠑⠦⡀⠀⠐⠀⠀⠀⠀⠀⠀⠀⠴⠀⠀⢀⡀⠀⠐⠉
⠀⠀⢸⠀⠀⠀⢰⠤⠀⠀⠀⠀⠀⠈⠑⠲⠋⠁⢠⠀⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⢀⠀⠲⠤⠄⢀⠠⠆
⠀⠀⠈⡄⠀⠀⢸⠁⠠⡀⠀⠀⠀⠀⠀⠀⠀⠀⢀⡀⠀⠀⠀⠀⠒⠉⠁⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⣠
⠀⠀⠀⠣⠀⠀⠜⠀⠀⠀⠐⠄⠀⣀⡠⠤⠀⠀⠀⠁⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠂⠈⠋⠈⠑
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠰⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢀⠀⠀⠀⠠⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⡠⠔⢂⠉⠄⠀⠠⠀⠀⠀⠀⠀⠄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠑⠀⠀⠀⠀⠀⢀⡀⠁⣀⡠⠭⠒⠂⠁⠀⠀⠀⠀⠀⠑⠀⠀⠀⠊
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠠⠖⠚⠉⠁
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠑⠂⠈

⠀⠀⠀⠀⠀⠀⠀⠰⣯⣧⣀⠀⠀⠀⠀⠀⠀⠀⠀⢀⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠻⣍⢧⡀⠀⠀⠀⠀⠀⣴⡿⠛⣿⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⢣⡙⢦⡀⠀⠀⠀⠙⠿⠿⢿⣷⠶⣶⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢦⡙⣦⠀⠀⠀⠀⠀⠸⣿⣤⣾⢇⣴⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢿⣷⠄⠀⠀⠀⠀⠈⠉⢡⣿⡋⣠⣶⢆⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠻⣦⠀⠀⠀⠀⠀⠀⠙⠿⠟⣱⣿⣻⣦
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠘⢿⣼⡿⣡⣾⠇
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⢷⣄⠀⠀⠀⠀⠀⠀⠈⠸⣿⣅⠀⣠⣶⣄⡀
⠀⠀⠀⠀⠀⠀⠀⠀⢠⣾⢷⡄⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠹⣦⡀⠀⠀⠀⠀⠀⠀⠈⠉⢾⣿⢿⡟⠃
⠀⠀⠀⠀⠀⠀⠀⠰⣷⣽⣷⠀⢹⣿⢷⢆⣄⠀⠀⠀⠀⠀⠀⠈⢹⣶⣄⠀⠀⠀⠀⠀⠀⠀⠙⠓
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠉⠁⠠⢿⠃⢀⣾⢏⣴⡶⣶⠄⣀⠀⠀⠀⠹⣏⠳⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠘⠛⠸⣿⣤⠀⣼⣿⣴⡶⠄⠀⠈⠳⣌⠳⡄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠘⠿⠛⣿⡀⠀⠀⠀⠀⠈⠳⡝⢦⡀
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠁⠀⠀⠀⠀⠀⠀⠙⢻⣿⢤
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠙⠉

⠀⠀⠀⢼⣇⠀⠀⠀⠀⣸⣷⣷⣷⠀⠀⠀⠀⠀⣼⣼⣧⣧⠀⠀⠀⠀⠀⣠⣶⣷⣦⡄⠀⠀⠀⠀⠀⠀⢀⢠⣰⣦⣧⣶⣠⣀
⠀⠀⠀⢺⡿⠀⠀⠀⠀⢿⣿⣿⣿⠀⠀⠀⠀⠈⣿⣿⣿⣿⠀⠀⠀⠀⠐⣿⣿⣿⣿⡿⠀⠀⠀⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⣿⣿
⠀⠀⠀⠘⠃⠀⠀⠀⠀⠘⠛⠛⠉⠀⠀⠀⠀⠀⠙⠛⠛⠋⠀⠀⠀⠀⠀⠙⠛⠛⠛⠃⠀⠀⠀⠀⠀⠘⠙⠛⠛⠋⠋⠛⠙⠙⠙
⠀⠀⢠⣶⣰⡆⠀⠀⠀⣴⣀⣆⣰⡀⠀⠀⠀⠀⢀⣦⣶⡀⠀⠀⡀⡀⣤⢠⣀⣆⣠⢠⣀⡀⡀⠀⠀⢰⣴⣆⣶⣶⣰⣰⣆⣦⣶⡀
⠀⠀⠸⣿⣿⡇⠀⠀⠀⣿⣿⣿⣿⡇⠀⠀⠀⠀⢸⣿⣿⡇⠀⠰⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⠇
⠀⠀⠀⠟⠿⠃⠀⠀⠀⠻⠘⠇⠻⠃⠀⠀⠀⠀⠀⠟⠿⠂⠀⠀⠁⠁⠋⠘⠛⠇⠛⠘⠉⠁⠁⠀⠀⠸⠛⠛⠿⠿⠿⠛⠟⠿⠿
⠀⠀⢠⣤⡄⡄⠀⠀⠀⣤⠀⡄⢠⡄⠀⠀⠀⠀⢠⢠⡄⡄⠀⠀⠀⠀⠀⢀⠀⡄⢀⠀⠀⠀⠀⠀⠀⢠⣠⡄⣤⣤⢠⣠⡄⣄⣤
⠀⠀⣿⣿⣿⣿⠀⠀⢰⣿⣾⣿⣿⡇⠀⠀⠀⠀⣿⣿⣿⣿⠀⢀⣦⣦⣿⣾⣾⣿⣾⣾⣶⣆⡆⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠹⠿⡿⡿⠀⠀⠈⠿⠸⠏⢹⠇⠀⠀⠀⠀⠽⠿⠿⡏⠀⠈⠟⠟⠿⠻⠿⠏⠿⠹⠿⠿⠇⠀⠀⢹⢿⠿⠿⣿⠿⠿⠿⡿⠿⠄
⠀⠀⢀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⢀⢠⡄⡀⡄⡀⣄⣀⣠
⠀⠀⣼⣧⣧⣴⣠⣀⠀⠀⠀⢠⣿⣼⣦⣧⣦⣦⠀⠀⠀⣀⢠⡀⣄⣶⣴⣼⣾⡆⠀⠀⢠⠀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢻⡟⣿⣿⢿⢿⠇⠀⠀⠘⣿⢻⡻⡟⡟⠟⠀⠀⢠⣿⣿⣿⣿⣿⣿⠿⠛⠀⠀⠀⢿⣿⡆⠀⠀⢻⣿⣿⣿⣿⣿⣿⣿⣿⣿⠁
⠀⠀⠈⠁⠁⠈⠈⠈⠀⠀⠀⠀⠉⠀⠀⠀⠀⠀⠀⠀⠈⣿⠻⠟⠃⠉⠀⠀⠀⠀⠀⠀⣿⣿⡇⠀⠀⠀⠈⠁⢈⡁⠀⡍⠀⠉
⠀⠀⢀⣆⣶⣴⣤⣤⣤⣤⠀⢰⣦⣦⣤⣄⣀⡀⠀⠀⠀⠀⢀⢀⡀⣄⣤⣦⣦⠀⠀⠀⢿⣿⡇⠀⠀⠀⠀⣼⣿⣿⣿⣿⣿
⠀⠀⢸⣿⣿⣿⣿⣿⣿⣿⠀⢺⣿⣿⣿⣿⣿⣿⡆⠀⢀⣿⣾⣾⣿⣿⣿⣿⡟⠀⠀⠀⢸⣿⡇⠀⠀⠀⠀⢻⣿⣿⣿⣿⣿⠁
⠀⠀⠘⠛⠛⠛⠉⠉⠉⠉⠀⠘⠛⠛⠛⠛⠛⠛⠀⠀⠘⣿⣿⡿⡟⠟⠉⠁⠀⠀⠀⠀⠀⠛⠀⠀⠀⠀⠀⠈⠙⠋⠙⠛⠉
⠀⠀⢀⡀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡀⠀⢀⠀⠀⠀⠙⠀⠀⠀⠀⠀⠀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⣴⣦⣴⣦⣤⣶⣤
⠀⠀⣾⣷⣴⣷⣦⣿⣶⣼⣶⣧⣦⣧⣶⣿⣶⣾⡆⠀⠀⠀⠀⣷⣷⣾⣼⣷⣷⣿⣼⣾⣷⡇⠀⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⢹⣿⣿⣿⣿⣿⣿⢿⣿⣿⣿⣿⣿⣿⣿⣿⠃⠀⠀⠀⠀⠋⠿⢿⢻⣿⣿⣿⢻⠟⠏⠃⠀⠀⠀⠀⠹⣿⣿⢿⣿⢿⣿⡿
⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢠⠀⢀⡀⠀⡄⠀⣀⠀⣀
⠀⠀⢰⡄⠀⣤⠀⠀⡄⠀⢀⢀⠀⢀⡀⠀⡄⠀⣤⠀⠀⠀⠀⣰⣴⣠⣠⣄⣆⣤⣠⣰⣴⡀⠀⠀⠀⣾⣿⣿⣿⣿⣿⣿⣿⣿⣿⡆
⠀⠀⣿⣿⣷⣿⣿⣾⣿⣿⣿⣿⣾⣿⣷⣿⣿⣿⣿⡆⠀⠀⠀⠹⢿⣿⣿⣿⣿⣿⣿⢿⠿⠁⠀⠀⠀⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⡇
⠀⠀⠸⠏⠏⠿⠹⠋⠿⠿⠻⠿⠽⠿⠿⠿⠿⠏⠿⠀⠀⠀⠀⠀⠀⠈⠹⠻⠿⠛⠉⠀⠀⠀⠀⠀⠀⠘⠻⠟⠻⠟⠻⠿⠛⠿⠛
⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀⣀
⣿⣿⠛⢻⡟⡛⢻⡿⣻⢿⡛⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿⣿

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣤⣄
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢀⣀⣀⣀⣀⣀⣀⣀⣀⣀⡀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⢀⡴⠶⣄⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⡼⠲⠶⠾⠀⠐⠶⠶⠶⠖⡆⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⣼⠁⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⣿⣶⡶⢲⠒⢶⣶⣖⣀⡈⣇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⣇⠀⠀⠀⠀⣰⣹⠀⠀⢸⠀⠀⠀⠀⠀⠀⣿
⠀⠀⠀⠀⠀⠀⠀⠀⢘⣇⣤⣤⣀⣀⣉⣉⠉⢹⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠛⠛⠛⠙⠉⣻⠀⠀⢸⡏⠋⠛⠋⠛⠙⠉
⠀⠀⠀⠀⠀⡤⠤⠤⠬⠧⠤⠴⠤⠤⠤⠤⠤⠾⠤⠤⠤⠤⠤⠤⠤⠴⠶⠶⠶⠶⠶⠶⠶⠶⠶⠤⠶⠿⠶⠶⠾⠷⠦⠾⠭⠵⠤⠤⢤
⠀⠀⠀⠀⠀⠓⠒⡖⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⠒⢲⠒⠚
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸
⠀⠀⠀⠀⠀⠀⠀⡟⡏⠉⡟⡏⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⠉⢹⢹⠉⢹⢻
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⡇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⡇⡇⠀⣇⡇⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⢸⢸⠀⢸⢸
⠀⠀⠀⠀⠀⠀⠀⠙⠃⠀⠉⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠈⠉⠀⠘⠋